[{"data":1,"prerenderedAt":13126},["ShallowReactive",2],{"blog-post-es-lanzando-una-moneda-hasta-obtener-pi":3,"locale-switcher-blog-es-lanzando-una-moneda-hasta-obtener-pi":13123,"blog-siblings-es-lanzando-una-moneda-hasta-obtener-pi":13125},{"id":4,"title":5,"_locale":6,"body":7,"date":12967,"description":12968,"extension":12969,"meta":12970,"navigation":13117,"path":13118,"published":13117,"seo":13119,"slug":13120,"stem":13121,"__hash__":13122},"blog\u002Fblog\u002Fpi-coin-flips-catalan-numbers.es.md","Lanzando una moneda hasta que obtenemos PI","es",{"type":8,"value":9,"toc":12951},"minimal",[10,14,175,214,223,226,248,251,558,563,579,584,587,628,848,852,859,904,1262,1266,1273,1280,1283,1293,1373,1499,1502,1508,1627,1664,1802,2150,2154,2161,2165,3543,3546,3969,4067,4070,4074,4212,4241,4244,4276,4549,4859,4862,5371,5878,6142,6149,6153,6355,6359,6418,7221,7280,7721,7724,9159,9332,9336,9339,9599,9864,10385,10388,11161,11168,11670,11759,11814,12182,12294,12434,12437,12745,12790,12793,12829,12835,12839,12842,12873,12905,12908,12940,12943],[11,12,5],"h1",{"id":13},"lanzando-una-moneda-hasta-que-obtenemos-pi",[15,16,17,18,25,26,31,32,174],"p",{},"Matt Parker, del canal ",[19,20,24],"a",{"href":21,"rel":22},"https:\u002F\u002Fwww.youtube.com\u002F@standupmaths",[23],"nofollow","Stand-up Maths",", publicó recientemente ",[19,27,30],{"href":28,"rel":29},"https:\u002F\u002Fwww.youtube.com\u002Fwatch?v=kahGSss6SsU",[23],"un vídeo"," con una premisa tan simple de enunciar como difícil de creer: si lanzamos una moneda repetidamente hasta que haya más caras que cruces, y calculamos la proporción media de caras respecto al total de lanzamientos... el resultado es ",[33,34,37,67],"span",{"className":35},[36],"katex",[33,38,41],{"className":39},[40],"katex-mathml",[42,43,45],"math",{"xmlns":44},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[46,47,48,62],"semantics",{},[49,50,51],"mrow",{},[52,53,54,58],"mfrac",{},[55,56,57],"mi",{},"π",[59,60,61],"mn",{},"4",[63,64,66],"annotation",{"encoding":65},"application\u002Fx-tex","\\frac{\\pi}{4}",[33,68,72],{"className":69,"ariaHidden":71},[70],"katex-html","true",[33,73,76,81],{"className":74},[75],"base",[33,77],{"className":78,"style":80},[79],"strut","height:1.0404em;vertical-align:-0.345em;",[33,82,85,90,170],{"className":83},[84],"mord",[33,86],{"className":87},[88,89],"mopen","nulldelimiter",[33,91,93],{"className":92},[52],[33,94,98,161],{"className":95},[96,97],"vlist-t","vlist-t2",[33,99,102,156],{"className":100},[101],"vlist-r",[33,103,107,128,139],{"className":104,"style":106},[105],"vlist","height:0.6954em;",[33,108,110,115],{"style":109},"top:-2.655em;",[33,111],{"className":112,"style":114},[113],"pstrut","height:3em;",[33,116,122],{"className":117},[118,119,120,121],"sizing","reset-size6","size3","mtight",[33,123,125],{"className":124},[84,121],[33,126,61],{"className":127},[84,121],[33,129,131,134],{"style":130},"top:-3.23em;",[33,132],{"className":133,"style":114},[113],[33,135],{"className":136,"style":138},[137],"frac-line","border-bottom-width:0.04em;",[33,140,142,145],{"style":141},"top:-3.394em;",[33,143],{"className":144,"style":114},[113],[33,146,148],{"className":147},[118,119,120,121],[33,149,151],{"className":150},[84,121],[33,152,57],{"className":153,"style":155},[84,154,121],"mathnormal","margin-right:0.0359em;",[33,157,160],{"className":158},[159],"vlist-s","​",[33,162,164],{"className":163},[101],[33,165,168],{"className":166,"style":167},[105],"height:0.345em;",[33,169],{},[33,171],{"className":172},[173,89],"mclose",".",[15,176,177,178,208,209,213],{},"Sí, ",[33,179,181,195],{"className":180},[36],[33,182,184],{"className":183},[40],[42,185,186],{"xmlns":44},[46,187,188,192],{},[49,189,190],{},[55,191,57],{},[63,193,194],{"encoding":65},"\\pi",[33,196,198],{"className":197,"ariaHidden":71},[70],[33,199,201,205],{"className":200},[75],[33,202],{"className":203,"style":204},[79],"height:0.4306em;",[33,206,57],{"className":207,"style":155},[84,154],". ",[210,211,212],"em",{},"El número de los círculos",". Escondido dentro de lanzamientos de moneda.",[15,215,216,217,222],{},"En nuestro ",[19,218,221],{"href":219,"rel":220},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Ftriangulo-de-tartaglia-combinatoria-y-probabilidad\u002F",[23],"artículo anterior sobre el triángulo de Tartaglia"," vimos cómo un triángulo de sumas sencillas contenía, en su interior, la combinatoria, el álgebra y la probabilidad. Hoy vamos a tirar del mismo hilo. Y nos va a llevar a un lugar que nadie esperaba.",[15,224,225],{},"Las reglas del experimento son sencillas:",[227,228,229,233,241],"ol",{},[230,231,232],"li",{},"Lanzamos una moneda justa repetidamente.",[230,234,235,236,240],{},"Nos detenemos ",[237,238,239],"strong",{},"en el momento exacto"," en que el número de caras supera al de cruces.",[230,242,243,244,247],{},"Anotamos la ",[237,245,246],{},"proporción de caras"," respecto al total de lanzamientos de esa secuencia.",[15,249,250],{},"La pregunta del artículo será: si repetimos este proceso infinitas veces y calculamos la media de todas esas proporciones... ¿qué obtenemos?",[15,252,253,254,383,384,483,484,528,529,557],{},"Antes de responder, pensemos en qué rango puede moverse esa proporción. En el mejor de los casos, el primer lanzamiento es cara: paramos inmediatamente y la proporción es ",[33,255,257,282],{"className":256},[36],[33,258,260],{"className":259},[40],[42,261,262],{"xmlns":44},[46,263,264,279],{},[49,265,266,273,277],{},[52,267,268,271],{},[59,269,270],{},"1",[59,272,270],{},[274,275,276],"mo",{},"=",[59,278,270],{},[63,280,281],{"encoding":65},"\\frac{1}{1} = 1",[33,283,285,373],{"className":284,"ariaHidden":71},[70],[33,286,288,292,361,366,370],{"className":287},[75],[33,289],{"className":290,"style":291},[79],"height:1.1901em;vertical-align:-0.345em;",[33,293,295,298,358],{"className":294},[84],[33,296],{"className":297},[88,89],[33,299,301],{"className":300},[52],[33,302,304,350],{"className":303},[96,97],[33,305,307,347],{"className":306},[101],[33,308,311,325,333],{"className":309,"style":310},[105],"height:0.8451em;",[33,312,313,316],{"style":109},[33,314],{"className":315,"style":114},[113],[33,317,319],{"className":318},[118,119,120,121],[33,320,322],{"className":321},[84,121],[33,323,270],{"className":324},[84,121],[33,326,327,330],{"style":130},[33,328],{"className":329,"style":114},[113],[33,331],{"className":332,"style":138},[137],[33,334,335,338],{"style":141},[33,336],{"className":337,"style":114},[113],[33,339,341],{"className":340},[118,119,120,121],[33,342,344],{"className":343},[84,121],[33,345,270],{"className":346},[84,121],[33,348,160],{"className":349},[159],[33,351,353],{"className":352},[101],[33,354,356],{"className":355,"style":167},[105],[33,357],{},[33,359],{"className":360},[173,89],[33,362],{"className":363,"style":365},[364],"mspace","margin-right:0.2778em;",[33,367,276],{"className":368},[369],"mrel",[33,371],{"className":372,"style":365},[364],[33,374,376,380],{"className":375},[75],[33,377],{"className":378,"style":379},[79],"height:0.6444em;",[33,381,270],{"className":382},[84]," (el 100 % son caras). En el peor, necesitamos miles de lanzamientos hasta que las caras superen a las cruces por uno, y la proporción será apenas superior a ",[33,385,387,406],{"className":386},[36],[33,388,390],{"className":389},[40],[42,391,392],{"xmlns":44},[46,393,394,403],{},[49,395,396],{},[52,397,398,400],{},[59,399,270],{},[59,401,402],{},"2",[63,404,405],{"encoding":65},"\\frac{1}{2}",[33,407,409],{"className":408,"ariaHidden":71},[70],[33,410,412,415],{"className":411},[75],[33,413],{"className":414,"style":291},[79],[33,416,418,421,480],{"className":417},[84],[33,419],{"className":420},[88,89],[33,422,424],{"className":423},[52],[33,425,427,472],{"className":426},[96,97],[33,428,430,469],{"className":429},[101],[33,431,433,447,455],{"className":432,"style":310},[105],[33,434,435,438],{"style":109},[33,436],{"className":437,"style":114},[113],[33,439,441],{"className":440},[118,119,120,121],[33,442,444],{"className":443},[84,121],[33,445,402],{"className":446},[84,121],[33,448,449,452],{"style":130},[33,450],{"className":451,"style":114},[113],[33,453],{"className":454,"style":138},[137],[33,456,457,460],{"style":141},[33,458],{"className":459,"style":114},[113],[33,461,463],{"className":462},[118,119,120,121],[33,464,466],{"className":465},[84,121],[33,467,270],{"className":468},[84,121],[33,470,160],{"className":471},[159],[33,473,475],{"className":474},[101],[33,476,478],{"className":477,"style":167},[105],[33,479],{},[33,481],{"className":482},[173,89],". Así que la media estará en algún punto entre ",[33,485,487,502],{"className":486},[36],[33,488,490],{"className":489},[40],[42,491,492],{"xmlns":44},[46,493,494,499],{},[49,495,496],{},[59,497,498],{},"0,5",[63,500,501],{"encoding":65},"0{,}5",[33,503,505],{"className":504,"ariaHidden":71},[70],[33,506,508,512,516,524],{"className":507},[75],[33,509],{"className":510,"style":511},[79],"height:0.8389em;vertical-align:-0.1944em;",[33,513,515],{"className":514},[84],"0",[33,517,519],{"className":518},[84],[33,520,523],{"className":521},[522],"mpunct",",",[33,525,527],{"className":526},[84],"5"," y ",[33,530,532,545],{"className":531},[36],[33,533,535],{"className":534},[40],[42,536,537],{"xmlns":44},[46,538,539,543],{},[49,540,541],{},[59,542,270],{},[63,544,270],{"encoding":65},[33,546,548],{"className":547,"ariaHidden":71},[70],[33,549,551,554],{"className":550},[75],[33,552],{"className":553,"style":379},[79],[33,555,270],{"className":556},[84],". Veamos exactamente dónde.",[559,560,562],"h2",{"id":561},"construyendo-las-secuencias-paso-a-paso","Construyendo las secuencias, paso a paso",[15,564,565,566,569,570,573,574,174],{},"Vamos a construir las secuencias posibles empezando por las más cortas. Usaremos ",[237,567,568],{},"O"," para cara y ",[237,571,572],{},"X"," para cruz — la misma notación que usamos en ",[19,575,578],{"href":576,"rel":577},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fplay\u002F",[23],"nuestra herramienta de lanzamiento",[580,581,583],"h3",{"id":582},"longitud-1","Longitud 1",[15,585,586],{},"La secuencia más corta posible: lanzamos una vez y sale cara.",[15,588,591],{"className":589},[590],"math-block",[33,592,594,610],{"className":593},[36],[33,595,597],{"className":596},[40],[42,598,599],{"xmlns":44},[46,600,601,607],{},[49,602,603],{},[604,605,568],"mtext",{"mathvariant":606},"bold",[63,608,609],{"encoding":65},"\\textbf{O}",[33,611,613],{"className":612,"ariaHidden":71},[70],[33,614,616,620],{"className":615},[75],[33,617],{"className":618,"style":619},[79],"height:0.6861em;",[33,621,624],{"className":622},[84,623],"text",[33,625,568],{"className":626},[84,627],"textbf",[15,629,630,631,750,751,174],{},"Paramos. Tenemos 1 cara y 0 cruces. La proporción es ",[33,632,634,655],{"className":633},[36],[33,635,637],{"className":636},[40],[42,638,639],{"xmlns":44},[46,640,641,653],{},[49,642,643,649,651],{},[52,644,645,647],{},[59,646,270],{},[59,648,270],{},[274,650,276],{},[59,652,270],{},[63,654,281],{"encoding":65},[33,656,658,741],{"className":657,"ariaHidden":71},[70],[33,659,661,664,732,735,738],{"className":660},[75],[33,662],{"className":663,"style":291},[79],[33,665,667,670,729],{"className":666},[84],[33,668],{"className":669},[88,89],[33,671,673],{"className":672},[52],[33,674,676,721],{"className":675},[96,97],[33,677,679,718],{"className":678},[101],[33,680,682,696,704],{"className":681,"style":310},[105],[33,683,684,687],{"style":109},[33,685],{"className":686,"style":114},[113],[33,688,690],{"className":689},[118,119,120,121],[33,691,693],{"className":692},[84,121],[33,694,270],{"className":695},[84,121],[33,697,698,701],{"style":130},[33,699],{"className":700,"style":114},[113],[33,702],{"className":703,"style":138},[137],[33,705,706,709],{"style":141},[33,707],{"className":708,"style":114},[113],[33,710,712],{"className":711},[118,119,120,121],[33,713,715],{"className":714},[84,121],[33,716,270],{"className":717},[84,121],[33,719,160],{"className":720},[159],[33,722,724],{"className":723},[101],[33,725,727],{"className":726,"style":167},[105],[33,728],{},[33,730],{"className":731},[173,89],[33,733],{"className":734,"style":365},[364],[33,736,276],{"className":737},[369],[33,739],{"className":740,"style":365},[364],[33,742,744,747],{"className":743},[75],[33,745],{"className":746,"style":379},[79],[33,748,270],{"className":749},[84]," y la probabilidad de que esto ocurra es ",[33,752,754,771],{"className":753},[36],[33,755,757],{"className":756},[40],[42,758,759],{"xmlns":44},[46,760,761,769],{},[49,762,763],{},[52,764,765,767],{},[59,766,270],{},[59,768,402],{},[63,770,405],{"encoding":65},[33,772,774],{"className":773,"ariaHidden":71},[70],[33,775,777,780],{"className":776},[75],[33,778],{"className":779,"style":291},[79],[33,781,783,786,845],{"className":782},[84],[33,784],{"className":785},[88,89],[33,787,789],{"className":788},[52],[33,790,792,837],{"className":791},[96,97],[33,793,795,834],{"className":794},[101],[33,796,798,812,820],{"className":797,"style":310},[105],[33,799,800,803],{"style":109},[33,801],{"className":802,"style":114},[113],[33,804,806],{"className":805},[118,119,120,121],[33,807,809],{"className":808},[84,121],[33,810,402],{"className":811},[84,121],[33,813,814,817],{"style":130},[33,815],{"className":816,"style":114},[113],[33,818],{"className":819,"style":138},[137],[33,821,822,825],{"style":141},[33,823],{"className":824,"style":114},[113],[33,826,828],{"className":827},[118,119,120,121],[33,829,831],{"className":830},[84,121],[33,832,270],{"className":833},[84,121],[33,835,160],{"className":836},[159],[33,838,840],{"className":839},[101],[33,841,843],{"className":842,"style":167},[105],[33,844],{},[33,846],{"className":847},[173,89],[580,849,851],{"id":850},"longitud-3","Longitud 3",[15,853,854,855,858],{},"Si el primer lanzamiento es cruz, no podemos parar. Y con sólo dos lanzamientos, lo mejor que podemos lograr es empatar (1 cara, 1 cruz) — empatar no basta, necesitamos que las caras ",[210,856,857],{},"superen"," a las cruces. Así que la siguiente secuencia posible tiene longitud 3:",[15,860,862],{"className":861},[590],[33,863,865,888],{"className":864},[36],[33,866,868],{"className":867},[40],[42,869,870],{"xmlns":44},[46,871,872,885],{},[49,873,874,876,879,881,883],{},[604,875,572],{"mathvariant":606},[604,877,878],{}," ",[604,880,568],{"mathvariant":606},[604,882,878],{},[604,884,568],{"mathvariant":606},[63,886,887],{"encoding":65},"\\textbf{X O O}",[33,889,891],{"className":890,"ariaHidden":71},[70],[33,892,894,897],{"className":893},[75],[33,895],{"className":896,"style":619},[79],[33,898,900],{"className":899},[84,623],[33,901,903],{"className":902},[84,627],"X O O",[15,905,906,907,1006,1007,1261],{},"Cruz, cara, cara. El tercer lanzamiento nos da 2 caras frente a 1 cruz. Proporción: ",[33,908,910,929],{"className":909},[36],[33,911,913],{"className":912},[40],[42,914,915],{"xmlns":44},[46,916,917,926],{},[49,918,919],{},[52,920,921,923],{},[59,922,402],{},[59,924,925],{},"3",[63,927,928],{"encoding":65},"\\frac{2}{3}",[33,930,932],{"className":931,"ariaHidden":71},[70],[33,933,935,938],{"className":934},[75],[33,936],{"className":937,"style":291},[79],[33,939,941,944,1003],{"className":940},[84],[33,942],{"className":943},[88,89],[33,945,947],{"className":946},[52],[33,948,950,995],{"className":949},[96,97],[33,951,953,992],{"className":952},[101],[33,954,956,970,978],{"className":955,"style":310},[105],[33,957,958,961],{"style":109},[33,959],{"className":960,"style":114},[113],[33,962,964],{"className":963},[118,119,120,121],[33,965,967],{"className":966},[84,121],[33,968,925],{"className":969},[84,121],[33,971,972,975],{"style":130},[33,973],{"className":974,"style":114},[113],[33,976],{"className":977,"style":138},[137],[33,979,980,983],{"style":141},[33,981],{"className":982,"style":114},[113],[33,984,986],{"className":985},[118,119,120,121],[33,987,989],{"className":988},[84,121],[33,990,402],{"className":991},[84,121],[33,993,160],{"className":994},[159],[33,996,998],{"className":997},[101],[33,999,1001],{"className":1000,"style":167},[105],[33,1002],{},[33,1004],{"className":1005},[173,89]," (2 caras de 3 lanzamientos). Probabilidad: ",[33,1008,1010,1050],{"className":1009},[36],[33,1011,1013],{"className":1012},[40],[42,1014,1015],{"xmlns":44},[46,1016,1017,1047],{},[49,1018,1019,1038,1040],{},[1020,1021,1022,1036],"msup",{},[49,1023,1024,1027,1033],{},[274,1025,1026],{"fence":71},"(",[52,1028,1029,1031],{},[59,1030,270],{},[59,1032,402],{},[274,1034,1035],{"fence":71},")",[59,1037,925],{},[274,1039,276],{},[52,1041,1042,1044],{},[59,1043,270],{},[59,1045,1046],{},"8",[63,1048,1049],{"encoding":65},"\\left(\\frac{1}{2}\\right)^3 = \\frac{1}{8}",[33,1051,1053,1187],{"className":1052,"ariaHidden":71},[70],[33,1054,1056,1060,1178,1181,1184],{"className":1055},[75],[33,1057],{"className":1058,"style":1059},[79],"height:1.404em;vertical-align:-0.35em;",[33,1061,1064,1151],{"className":1062},[1063],"minner",[33,1065,1067,1077,1145],{"className":1066},[1063],[33,1068,1072],{"className":1069,"style":1071},[88,1070],"delimcenter","top:0em;",[33,1073,1026],{"className":1074},[1075,1076],"delimsizing","size1",[33,1078,1080,1083,1142],{"className":1079},[84],[33,1081],{"className":1082},[88,89],[33,1084,1086],{"className":1085},[52],[33,1087,1089,1134],{"className":1088},[96,97],[33,1090,1092,1131],{"className":1091},[101],[33,1093,1095,1109,1117],{"className":1094,"style":310},[105],[33,1096,1097,1100],{"style":109},[33,1098],{"className":1099,"style":114},[113],[33,1101,1103],{"className":1102},[118,119,120,121],[33,1104,1106],{"className":1105},[84,121],[33,1107,402],{"className":1108},[84,121],[33,1110,1111,1114],{"style":130},[33,1112],{"className":1113,"style":114},[113],[33,1115],{"className":1116,"style":138},[137],[33,1118,1119,1122],{"style":141},[33,1120],{"className":1121,"style":114},[113],[33,1123,1125],{"className":1124},[118,119,120,121],[33,1126,1128],{"className":1127},[84,121],[33,1129,270],{"className":1130},[84,121],[33,1132,160],{"className":1133},[159],[33,1135,1137],{"className":1136},[101],[33,1138,1140],{"className":1139,"style":167},[105],[33,1141],{},[33,1143],{"className":1144},[173,89],[33,1146,1148],{"className":1147,"style":1071},[173,1070],[33,1149,1035],{"className":1150},[1075,1076],[33,1152,1155],{"className":1153},[1154],"msupsub",[33,1156,1158],{"className":1157},[96],[33,1159,1161],{"className":1160},[101],[33,1162,1165],{"className":1163,"style":1164},[105],"height:1.054em;",[33,1166,1168,1172],{"style":1167},"top:-3.3029em;margin-right:0.05em;",[33,1169],{"className":1170,"style":1171},[113],"height:2.7em;",[33,1173,1175],{"className":1174},[118,119,120,121],[33,1176,925],{"className":1177},[84,121],[33,1179],{"className":1180,"style":365},[364],[33,1182,276],{"className":1183},[369],[33,1185],{"className":1186,"style":365},[364],[33,1188,1190,1193],{"className":1189},[75],[33,1191],{"className":1192,"style":291},[79],[33,1194,1196,1199,1258],{"className":1195},[84],[33,1197],{"className":1198},[88,89],[33,1200,1202],{"className":1201},[52],[33,1203,1205,1250],{"className":1204},[96,97],[33,1206,1208,1247],{"className":1207},[101],[33,1209,1211,1225,1233],{"className":1210,"style":310},[105],[33,1212,1213,1216],{"style":109},[33,1214],{"className":1215,"style":114},[113],[33,1217,1219],{"className":1218},[118,119,120,121],[33,1220,1222],{"className":1221},[84,121],[33,1223,1046],{"className":1224},[84,121],[33,1226,1227,1230],{"style":130},[33,1228],{"className":1229,"style":114},[113],[33,1231],{"className":1232,"style":138},[137],[33,1234,1235,1238],{"style":141},[33,1236],{"className":1237,"style":114},[113],[33,1239,1241],{"className":1240},[118,119,120,121],[33,1242,1244],{"className":1243},[84,121],[33,1245,270],{"className":1246},[84,121],[33,1248,160],{"className":1249},[159],[33,1251,1253],{"className":1252},[101],[33,1254,1256],{"className":1255,"style":167},[105],[33,1257],{},[33,1259],{"className":1260},[173,89],". Y es la única secuencia posible de esta longitud.",[580,1263,1265],{"id":1264},"longitud-5","Longitud 5",[15,1267,1268,1269,1272],{},"Aquí se pone interesante. Necesitamos 3 caras y 2 cruces, y el quinto lanzamiento (el que nos detiene) tiene que ser cara. Además — y esto es clave — ",[237,1270,1271],{},"en ningún momento antes del final podemos haber tenido más caras que cruces",", porque si eso hubiera ocurrido, nos habríamos detenido antes.",[15,1274,1275,1276,1279],{},"No basta con que la secuencia termine con más caras que cruces; tiene que ser ",[237,1277,1278],{},"la primera vez"," que ocurre.",[15,1281,1282],{},"¿Cuántas secuencias cumplen estas condiciones? Sólo dos:",[1284,1285,1290],"pre",{"className":1286,"code":1288,"language":623,"meta":1289},[1287],"language-text","X X O O O\nX O X O O\n","",[1291,1292,1288],"code",{"__ignoreMap":1289},[15,1294,1295,1296,1299,1300,1336,1337,1372],{},"Comprobemos la primera: ",[237,1297,1298],{},"XXOOO",". Si contamos O como ",[33,1301,1303,1320],{"className":1302},[36],[33,1304,1306],{"className":1305},[40],[42,1307,1308],{"xmlns":44},[46,1309,1310,1317],{},[49,1311,1312,1315],{},[274,1313,1314],{},"+",[59,1316,270],{},[63,1318,1319],{"encoding":65},"+1",[33,1321,1323],{"className":1322,"ariaHidden":71},[70],[33,1324,1326,1330,1333],{"className":1325},[75],[33,1327],{"className":1328,"style":1329},[79],"height:0.7278em;vertical-align:-0.0833em;",[33,1331,1314],{"className":1332},[84],[33,1334,270],{"className":1335},[84]," y X como ",[33,1338,1340,1357],{"className":1339},[36],[33,1341,1343],{"className":1342},[40],[42,1344,1345],{"xmlns":44},[46,1346,1347,1354],{},[49,1348,1349,1352],{},[274,1350,1351],{},"−",[59,1353,270],{},[63,1355,1356],{"encoding":65},"-1",[33,1358,1360],{"className":1359,"ariaHidden":71},[70],[33,1361,1363,1366,1369],{"className":1362},[75],[33,1364],{"className":1365,"style":1329},[79],[33,1367,1351],{"className":1368},[84],[33,1370,270],{"className":1371},[84],", las sumas parciales son:",[15,1374,1376],{"className":1375},[590],[33,1377,1379,1426],{"className":1378},[36],[33,1380,1382],{"className":1381},[40],[42,1383,1384],{"xmlns":44},[46,1385,1386,1423],{},[49,1387,1388,1390,1392,1394,1397,1399,1401,1403,1405,1407,1409,1411,1413,1415,1417,1419,1421],{},[274,1389,1351],{},[59,1391,270],{},[274,1393,523],{"separator":71},[604,1395,1396],{},"  ",[274,1398,1351],{},[59,1400,402],{},[274,1402,523],{"separator":71},[604,1404,1396],{},[274,1406,1351],{},[59,1408,270],{},[274,1410,523],{"separator":71},[604,1412,1396],{},[59,1414,515],{},[274,1416,523],{"separator":71},[604,1418,1396],{},[274,1420,1314],{},[59,1422,270],{},[63,1424,1425],{"encoding":65},"-1,\\; -2,\\; -1,\\; 0,\\; +1",[33,1427,1429],{"className":1428,"ariaHidden":71},[70],[33,1430,1432,1435,1438,1441,1444,1447,1451,1454,1457,1460,1463,1466,1469,1472,1475,1478,1481,1484,1487,1490,1493,1496],{"className":1431},[75],[33,1433],{"className":1434,"style":511},[79],[33,1436,1351],{"className":1437},[84],[33,1439,270],{"className":1440},[84],[33,1442,523],{"className":1443},[522],[33,1445],{"className":1446,"style":365},[364],[33,1448],{"className":1449,"style":1450},[364],"margin-right:0.1667em;",[33,1452,1351],{"className":1453},[84],[33,1455,402],{"className":1456},[84],[33,1458,523],{"className":1459},[522],[33,1461],{"className":1462,"style":365},[364],[33,1464],{"className":1465,"style":1450},[364],[33,1467,1351],{"className":1468},[84],[33,1470,270],{"className":1471},[84],[33,1473,523],{"className":1474},[522],[33,1476],{"className":1477,"style":365},[364],[33,1479],{"className":1480,"style":1450},[364],[33,1482,515],{"className":1483},[84],[33,1485,523],{"className":1486},[522],[33,1488],{"className":1489,"style":365},[364],[33,1491],{"className":1492,"style":1450},[364],[33,1494,1314],{"className":1495},[84],[33,1497,270],{"className":1498},[84],[15,1500,1501],{},"Nunca supera el cero hasta el quinto lanzamiento. ✅",[15,1503,1504,1505,174],{},"Ahora la segunda: ",[237,1506,1507],{},"XOXOO",[15,1509,1511],{"className":1510},[590],[33,1512,1514,1558],{"className":1513},[36],[33,1515,1517],{"className":1516},[40],[42,1518,1519],{"xmlns":44},[46,1520,1521,1555],{},[49,1522,1523,1525,1527,1529,1531,1533,1535,1537,1539,1541,1543,1545,1547,1549,1551,1553],{},[274,1524,1351],{},[59,1526,270],{},[274,1528,523],{"separator":71},[604,1530,1396],{},[59,1532,515],{},[274,1534,523],{"separator":71},[604,1536,1396],{},[274,1538,1351],{},[59,1540,270],{},[274,1542,523],{"separator":71},[604,1544,1396],{},[59,1546,515],{},[274,1548,523],{"separator":71},[604,1550,1396],{},[274,1552,1314],{},[59,1554,270],{},[63,1556,1557],{"encoding":65},"-1,\\; 0,\\; -1,\\; 0,\\; +1",[33,1559,1561],{"className":1560,"ariaHidden":71},[70],[33,1562,1564,1567,1570,1573,1576,1579,1582,1585,1588,1591,1594,1597,1600,1603,1606,1609,1612,1615,1618,1621,1624],{"className":1563},[75],[33,1565],{"className":1566,"style":511},[79],[33,1568,1351],{"className":1569},[84],[33,1571,270],{"className":1572},[84],[33,1574,523],{"className":1575},[522],[33,1577],{"className":1578,"style":365},[364],[33,1580],{"className":1581,"style":1450},[364],[33,1583,515],{"className":1584},[84],[33,1586,523],{"className":1587},[522],[33,1589],{"className":1590,"style":365},[364],[33,1592],{"className":1593,"style":1450},[364],[33,1595,1351],{"className":1596},[84],[33,1598,270],{"className":1599},[84],[33,1601,523],{"className":1602},[522],[33,1604],{"className":1605,"style":365},[364],[33,1607],{"className":1608,"style":1450},[364],[33,1610,515],{"className":1611},[84],[33,1613,523],{"className":1614},[522],[33,1616],{"className":1617,"style":365},[364],[33,1619],{"className":1620,"style":1450},[364],[33,1622,1314],{"className":1623},[84],[33,1625,270],{"className":1626},[84],[15,1628,1629,1630,1663],{},"Vuelve a cero en el segundo y cuarto lanzamiento, pero nunca lo supera. Es la primera vez que llega a ",[33,1631,1633,1648],{"className":1632},[36],[33,1634,1636],{"className":1635},[40],[42,1637,1638],{"xmlns":44},[46,1639,1640,1646],{},[49,1641,1642,1644],{},[274,1643,1314],{},[59,1645,270],{},[63,1647,1319],{"encoding":65},[33,1649,1651],{"className":1650,"ariaHidden":71},[70],[33,1652,1654,1657,1660],{"className":1653},[75],[33,1655],{"className":1656,"style":1329},[79],[33,1658,1314],{"className":1659},[84],[33,1661,270],{"className":1662},[84],". ✅",[15,1665,1666,1667,1670,1671,1767,1768,1801],{},"¿Y qué pasa con ",[237,1668,1669],{},"OXXOO","? Veamos: ",[33,1672,1674,1710],{"className":1673},[36],[33,1675,1677],{"className":1676},[40],[42,1678,1679],{"xmlns":44},[46,1680,1681,1707],{},[49,1682,1683,1685,1687,1689,1691,1693,1695,1697,1699,1701,1703,1705],{},[274,1684,1314],{},[59,1686,270],{},[274,1688,523],{"separator":71},[59,1690,515],{},[274,1692,523],{"separator":71},[274,1694,1351],{},[59,1696,270],{},[274,1698,523],{"separator":71},[59,1700,515],{},[274,1702,523],{"separator":71},[274,1704,1314],{},[59,1706,270],{},[63,1708,1709],{"encoding":65},"+1, 0, -1, 0, +1",[33,1711,1713],{"className":1712,"ariaHidden":71},[70],[33,1714,1716,1719,1722,1725,1728,1731,1734,1737,1740,1743,1746,1749,1752,1755,1758,1761,1764],{"className":1715},[75],[33,1717],{"className":1718,"style":511},[79],[33,1720,1314],{"className":1721},[84],[33,1723,270],{"className":1724},[84],[33,1726,523],{"className":1727},[522],[33,1729],{"className":1730,"style":1450},[364],[33,1732,515],{"className":1733},[84],[33,1735,523],{"className":1736},[522],[33,1738],{"className":1739,"style":1450},[364],[33,1741,1351],{"className":1742},[84],[33,1744,270],{"className":1745},[84],[33,1747,523],{"className":1748},[522],[33,1750],{"className":1751,"style":1450},[364],[33,1753,515],{"className":1754},[84],[33,1756,523],{"className":1757},[522],[33,1759],{"className":1760,"style":1450},[364],[33,1762,1314],{"className":1763},[84],[33,1765,270],{"className":1766},[84],". El primer lanzamiento ya nos pone en ",[33,1769,1771,1786],{"className":1770},[36],[33,1772,1774],{"className":1773},[40],[42,1775,1776],{"xmlns":44},[46,1777,1778,1784],{},[49,1779,1780,1782],{},[274,1781,1314],{},[59,1783,270],{},[63,1785,1319],{"encoding":65},[33,1787,1789],{"className":1788,"ariaHidden":71},[70],[33,1790,1792,1795,1798],{"className":1791},[75],[33,1793],{"className":1794,"style":1329},[79],[33,1796,1314],{"className":1797},[84],[33,1799,270],{"className":1800},[84],": nos habríamos detenido ahí. No es válida. ❌",[15,1803,1804,1805,1808,1809,2050,2051,2149],{},"Así que hay exactamente ",[237,1806,1807],{},"2 secuencias"," de longitud 5, cada una con probabilidad ",[33,1810,1812,1849],{"className":1811},[36],[33,1813,1815],{"className":1814},[40],[42,1816,1817],{"xmlns":44},[46,1818,1819,1846],{},[49,1820,1821,1837,1839],{},[1020,1822,1823,1835],{},[49,1824,1825,1827,1833],{},[274,1826,1026],{"fence":71},[52,1828,1829,1831],{},[59,1830,270],{},[59,1832,402],{},[274,1834,1035],{"fence":71},[59,1836,527],{},[274,1838,276],{},[52,1840,1841,1843],{},[59,1842,270],{},[59,1844,1845],{},"32",[63,1847,1848],{"encoding":65},"\\left(\\frac{1}{2}\\right)^5 = \\frac{1}{32}",[33,1850,1852,1976],{"className":1851,"ariaHidden":71},[70],[33,1853,1855,1858,1967,1970,1973],{"className":1854},[75],[33,1856],{"className":1857,"style":1059},[79],[33,1859,1861,1944],{"className":1860},[1063],[33,1862,1864,1870,1938],{"className":1863},[1063],[33,1865,1867],{"className":1866,"style":1071},[88,1070],[33,1868,1026],{"className":1869},[1075,1076],[33,1871,1873,1876,1935],{"className":1872},[84],[33,1874],{"className":1875},[88,89],[33,1877,1879],{"className":1878},[52],[33,1880,1882,1927],{"className":1881},[96,97],[33,1883,1885,1924],{"className":1884},[101],[33,1886,1888,1902,1910],{"className":1887,"style":310},[105],[33,1889,1890,1893],{"style":109},[33,1891],{"className":1892,"style":114},[113],[33,1894,1896],{"className":1895},[118,119,120,121],[33,1897,1899],{"className":1898},[84,121],[33,1900,402],{"className":1901},[84,121],[33,1903,1904,1907],{"style":130},[33,1905],{"className":1906,"style":114},[113],[33,1908],{"className":1909,"style":138},[137],[33,1911,1912,1915],{"style":141},[33,1913],{"className":1914,"style":114},[113],[33,1916,1918],{"className":1917},[118,119,120,121],[33,1919,1921],{"className":1920},[84,121],[33,1922,270],{"className":1923},[84,121],[33,1925,160],{"className":1926},[159],[33,1928,1930],{"className":1929},[101],[33,1931,1933],{"className":1932,"style":167},[105],[33,1934],{},[33,1936],{"className":1937},[173,89],[33,1939,1941],{"className":1940,"style":1071},[173,1070],[33,1942,1035],{"className":1943},[1075,1076],[33,1945,1947],{"className":1946},[1154],[33,1948,1950],{"className":1949},[96],[33,1951,1953],{"className":1952},[101],[33,1954,1956],{"className":1955,"style":1164},[105],[33,1957,1958,1961],{"style":1167},[33,1959],{"className":1960,"style":1171},[113],[33,1962,1964],{"className":1963},[118,119,120,121],[33,1965,527],{"className":1966},[84,121],[33,1968],{"className":1969,"style":365},[364],[33,1971,276],{"className":1972},[369],[33,1974],{"className":1975,"style":365},[364],[33,1977,1979,1982],{"className":1978},[75],[33,1980],{"className":1981,"style":291},[79],[33,1983,1985,1988,2047],{"className":1984},[84],[33,1986],{"className":1987},[88,89],[33,1989,1991],{"className":1990},[52],[33,1992,1994,2039],{"className":1993},[96,97],[33,1995,1997,2036],{"className":1996},[101],[33,1998,2000,2014,2022],{"className":1999,"style":310},[105],[33,2001,2002,2005],{"style":109},[33,2003],{"className":2004,"style":114},[113],[33,2006,2008],{"className":2007},[118,119,120,121],[33,2009,2011],{"className":2010},[84,121],[33,2012,1845],{"className":2013},[84,121],[33,2015,2016,2019],{"style":130},[33,2017],{"className":2018,"style":114},[113],[33,2020],{"className":2021,"style":138},[137],[33,2023,2024,2027],{"style":141},[33,2025],{"className":2026,"style":114},[113],[33,2028,2030],{"className":2029},[118,119,120,121],[33,2031,2033],{"className":2032},[84,121],[33,2034,270],{"className":2035},[84,121],[33,2037,160],{"className":2038},[159],[33,2040,2042],{"className":2041},[101],[33,2043,2045],{"className":2044,"style":167},[105],[33,2046],{},[33,2048],{"className":2049},[173,89]," y proporción ",[33,2052,2054,2072],{"className":2053},[36],[33,2055,2057],{"className":2056},[40],[42,2058,2059],{"xmlns":44},[46,2060,2061,2069],{},[49,2062,2063],{},[52,2064,2065,2067],{},[59,2066,925],{},[59,2068,527],{},[63,2070,2071],{"encoding":65},"\\frac{3}{5}",[33,2073,2075],{"className":2074,"ariaHidden":71},[70],[33,2076,2078,2081],{"className":2077},[75],[33,2079],{"className":2080,"style":291},[79],[33,2082,2084,2087,2146],{"className":2083},[84],[33,2085],{"className":2086},[88,89],[33,2088,2090],{"className":2089},[52],[33,2091,2093,2138],{"className":2092},[96,97],[33,2094,2096,2135],{"className":2095},[101],[33,2097,2099,2113,2121],{"className":2098,"style":310},[105],[33,2100,2101,2104],{"style":109},[33,2102],{"className":2103,"style":114},[113],[33,2105,2107],{"className":2106},[118,119,120,121],[33,2108,2110],{"className":2109},[84,121],[33,2111,527],{"className":2112},[84,121],[33,2114,2115,2118],{"style":130},[33,2116],{"className":2117,"style":114},[113],[33,2119],{"className":2120,"style":138},[137],[33,2122,2123,2126],{"style":141},[33,2124],{"className":2125,"style":114},[113],[33,2127,2129],{"className":2128},[118,119,120,121],[33,2130,2132],{"className":2131},[84,121],[33,2133,925],{"className":2134},[84,121],[33,2136,160],{"className":2137},[159],[33,2139,2141],{"className":2140},[101],[33,2142,2144],{"className":2143,"style":167},[105],[33,2145],{},[33,2147],{"className":2148},[173,89]," (3 caras de 5 lanzamientos).",[580,2151,2153],{"id":2152},"longitud-7","Longitud 7",[15,2155,2156,2157,2160],{},"El mismo razonamiento se aplica a secuencias de longitud 7 (4 caras, 3 cruces). Si las enumeramos todas — y créeme, lleva un rato — encontramos exactamente ",[237,2158,2159],{},"5 secuencias"," válidas.",[580,2162,2164],{"id":2163},"la-tabla-resumen","La tabla resumen",[2166,2167,2168,2247],"table",{},[2169,2170,2171],"thead",{},[2172,2173,2174,2238,2241,2244],"tr",{},[2175,2176,2178,2179,1035],"th",{"align":2177},"center","Longitud (",[33,2180,2182,2203],{"className":2181},[36],[33,2183,2185],{"className":2184},[40],[42,2186,2187],{"xmlns":44},[46,2188,2189,2200],{},[49,2190,2191,2193,2196,2198],{},[59,2192,402],{},[55,2194,2195],{},"n",[274,2197,1314],{},[59,2199,270],{},[63,2201,2202],{"encoding":65},"2n+1",[33,2204,2206,2229],{"className":2205,"ariaHidden":71},[70],[33,2207,2209,2212,2215,2218,2222,2226],{"className":2208},[75],[33,2210],{"className":2211,"style":1329},[79],[33,2213,402],{"className":2214},[84],[33,2216,2195],{"className":2217},[84,154],[33,2219],{"className":2220,"style":2221},[364],"margin-right:0.2222em;",[33,2223,1314],{"className":2224},[2225],"mbin",[33,2227],{"className":2228,"style":2221},[364],[33,2230,2232,2235],{"className":2231},[75],[33,2233],{"className":2234,"style":379},[79],[33,2236,270],{"className":2237},[84],[2175,2239,2240],{"align":2177},"Proporción de caras",[2175,2242,2243],{"align":2177},"Secuencias válidas",[2175,2245,2246],{"align":2177},"Probabilidad de cada una",[2248,2249,2250,2508,2765,3022,3282],"tbody",{},[2172,2251,2252,2307,2407,2409],{},[2253,2254,2255,2256,1035],"td",{"align":2177},"1 (",[33,2257,2259,2277],{"className":2258},[36],[33,2260,2262],{"className":2261},[40],[42,2263,2264],{"xmlns":44},[46,2265,2266,2274],{},[49,2267,2268,2270,2272],{},[55,2269,2195],{},[274,2271,276],{},[59,2273,515],{},[63,2275,2276],{"encoding":65},"n=0",[33,2278,2280,2298],{"className":2279,"ariaHidden":71},[70],[33,2281,2283,2286,2289,2292,2295],{"className":2282},[75],[33,2284],{"className":2285,"style":204},[79],[33,2287,2195],{"className":2288},[84,154],[33,2290],{"className":2291,"style":365},[364],[33,2293,276],{"className":2294},[369],[33,2296],{"className":2297,"style":365},[364],[33,2299,2301,2304],{"className":2300},[75],[33,2302],{"className":2303,"style":379},[79],[33,2305,515],{"className":2306},[84],[2253,2308,2309],{"align":2177},[33,2310,2312,2330],{"className":2311},[36],[33,2313,2315],{"className":2314},[40],[42,2316,2317],{"xmlns":44},[46,2318,2319,2327],{},[49,2320,2321],{},[52,2322,2323,2325],{},[59,2324,270],{},[59,2326,270],{},[63,2328,2329],{"encoding":65},"\\frac{1}{1}",[33,2331,2333],{"className":2332,"ariaHidden":71},[70],[33,2334,2336,2339],{"className":2335},[75],[33,2337],{"className":2338,"style":291},[79],[33,2340,2342,2345,2404],{"className":2341},[84],[33,2343],{"className":2344},[88,89],[33,2346,2348],{"className":2347},[52],[33,2349,2351,2396],{"className":2350},[96,97],[33,2352,2354,2393],{"className":2353},[101],[33,2355,2357,2371,2379],{"className":2356,"style":310},[105],[33,2358,2359,2362],{"style":109},[33,2360],{"className":2361,"style":114},[113],[33,2363,2365],{"className":2364},[118,119,120,121],[33,2366,2368],{"className":2367},[84,121],[33,2369,270],{"className":2370},[84,121],[33,2372,2373,2376],{"style":130},[33,2374],{"className":2375,"style":114},[113],[33,2377],{"className":2378,"style":138},[137],[33,2380,2381,2384],{"style":141},[33,2382],{"className":2383,"style":114},[113],[33,2385,2387],{"className":2386},[118,119,120,121],[33,2388,2390],{"className":2389},[84,121],[33,2391,270],{"className":2392},[84,121],[33,2394,160],{"className":2395},[159],[33,2397,2399],{"className":2398},[101],[33,2400,2402],{"className":2401,"style":167},[105],[33,2403],{},[33,2405],{"className":2406},[173,89],[2253,2408,270],{"align":2177},[2253,2410,2411],{"align":2177},[33,2412,2414,2431],{"className":2413},[36],[33,2415,2417],{"className":2416},[40],[42,2418,2419],{"xmlns":44},[46,2420,2421,2429],{},[49,2422,2423],{},[52,2424,2425,2427],{},[59,2426,270],{},[59,2428,402],{},[63,2430,405],{"encoding":65},[33,2432,2434],{"className":2433,"ariaHidden":71},[70],[33,2435,2437,2440],{"className":2436},[75],[33,2438],{"className":2439,"style":291},[79],[33,2441,2443,2446,2505],{"className":2442},[84],[33,2444],{"className":2445},[88,89],[33,2447,2449],{"className":2448},[52],[33,2450,2452,2497],{"className":2451},[96,97],[33,2453,2455,2494],{"className":2454},[101],[33,2456,2458,2472,2480],{"className":2457,"style":310},[105],[33,2459,2460,2463],{"style":109},[33,2461],{"className":2462,"style":114},[113],[33,2464,2466],{"className":2465},[118,119,120,121],[33,2467,2469],{"className":2468},[84,121],[33,2470,402],{"className":2471},[84,121],[33,2473,2474,2477],{"style":130},[33,2475],{"className":2476,"style":114},[113],[33,2478],{"className":2479,"style":138},[137],[33,2481,2482,2485],{"style":141},[33,2483],{"className":2484,"style":114},[113],[33,2486,2488],{"className":2487},[118,119,120,121],[33,2489,2491],{"className":2490},[84,121],[33,2492,270],{"className":2493},[84,121],[33,2495,160],{"className":2496},[159],[33,2498,2500],{"className":2499},[101],[33,2501,2503],{"className":2502,"style":167},[105],[33,2504],{},[33,2506],{"className":2507},[173,89],[2172,2509,2510,2564,2663,2665],{},[2253,2511,2512,2513,1035],{"align":2177},"3 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(",[33,2771,2773,2791],{"className":2772},[36],[33,2774,2776],{"className":2775},[40],[42,2777,2778],{"xmlns":44},[46,2779,2780,2788],{},[49,2781,2782,2784,2786],{},[55,2783,2195],{},[274,2785,276],{},[59,2787,402],{},[63,2789,2790],{"encoding":65},"n=2",[33,2792,2794,2812],{"className":2793,"ariaHidden":71},[70],[33,2795,2797,2800,2803,2806,2809],{"className":2796},[75],[33,2798],{"className":2799,"style":204},[79],[33,2801,2195],{"className":2802},[84,154],[33,2804],{"className":2805,"style":365},[364],[33,2807,276],{"className":2808},[369],[33,2810],{"className":2811,"style":365},[364],[33,2813,2815,2818],{"className":2814},[75],[33,2816],{"className":2817,"style":379},[79],[33,2819,402],{"className":2820},[84],[2253,2822,2823],{"align":2177},[33,2824,2826,2843],{"className":2825},[36],[33,2827,2829],{"className":2828},[40],[42,2830,2831],{"xmlns":44},[46,2832,2833,2841],{},[49,2834,2835],{},[52,2836,2837,2839],{},[59,2838,925],{},[59,2840,527],{},[63,2842,2071],{"encoding":65},[33,2844,2846],{"className":2845,"ariaHidden":71},[70],[33,2847,2849,2852],{"className":2848},[75],[33,2850],{"className":2851,"style":291},[79],[33,2853,2855,2858,2917],{"className":2854},[84],[33,2856],{"className":2857},[88,89],[33,2859,2861],{"className":2860},[52],[33,2862,2864,2909],{"className":2863},[96,97],[33,2865,2867,2906],{"className":2866},[101],[33,2868,2870,2884,2892],{"className":2869,"style":310},[105],[33,2871,2872,2875],{"style":109},[33,2873],{"className":2874,"style":114},[113],[33,2876,2878],{"className":2877},[118,119,120,121],[33,2879,2881],{"className":2880},[84,121],[33,2882,527],{"className":2883},[84,121],[33,2885,2886,2889],{"style":130},[33,2887],{"className":2888,"style":114},[113],[33,2890],{"className":2891,"style":138},[137],[33,2893,2894,2897],{"style":141},[33,2895],{"className":2896,"style":114},[113],[33,2898,2900],{"className":2899},[118,119,120,121],[33,2901,2903],{"className":2902},[84,121],[33,2904,925],{"className":2905},[84,121],[33,2907,160],{"className":2908},[159],[33,2910,2912],{"className":2911},[101],[33,2913,2915],{"className":2914,"style":167},[105],[33,2916],{},[33,2918],{"className":2919},[173,89],[2253,2921,402],{"align":2177},[2253,2923,2924],{"align":2177},[33,2925,2927,2945],{"className":2926},[36],[33,2928,2930],{"className":2929},[40],[42,2931,2932],{"xmlns":44},[46,2933,2934,2942],{},[49,2935,2936],{},[52,2937,2938,2940],{},[59,2939,270],{},[59,2941,1845],{},[63,2943,2944],{"encoding":65},"\\frac{1}{32}",[33,2946,2948],{"className":2947,"ariaHidden":71},[70],[33,2949,2951,2954],{"className":2950},[75],[33,2952],{"className":2953,"style":291},[79],[33,2955,2957,2960,3019],{"className":2956},[84],[33,2958],{"className":2959},[88,89],[33,2961,2963],{"className":2962},[52],[33,2964,2966,3011],{"className":2965},[96,97],[33,2967,2969,3008],{"className":2968},[101],[33,2970,2972,2986,2994],{"className":2971,"style":310},[105],[33,2973,2974,2977],{"style":109},[33,2975],{"className":2976,"style":114},[113],[33,2978,2980],{"className":2979},[118,119,120,121],[33,2981,2983],{"className":2982},[84,121],[33,2984,1845],{"className":2985},[84,121],[33,2987,2988,2991],{"style":130},[33,2989],{"className":2990,"style":114},[113],[33,2992],{"className":2993,"style":138},[137],[33,2995,2996,2999],{"style":141},[33,2997],{"className":2998,"style":114},[113],[33,3000,3002],{"className":3001},[118,119,120,121],[33,3003,3005],{"className":3004},[84,121],[33,3006,270],{"className":3007},[84,121],[33,3009,160],{"className":3010},[159],[33,3012,3014],{"className":3013},[101],[33,3015,3017],{"className":3016,"style":167},[105],[33,3018],{},[33,3020],{"className":3021},[173,89],[2172,3023,3024,3078,3179,3181],{},[2253,3025,3026,3027,1035],{"align":2177},"7 (",[33,3028,3030,3048],{"className":3029},[36],[33,3031,3033],{"className":3032},[40],[42,3034,3035],{"xmlns":44},[46,3036,3037,3045],{},[49,3038,3039,3041,3043],{},[55,3040,2195],{},[274,3042,276],{},[59,3044,925],{},[63,3046,3047],{"encoding":65},"n=3",[33,3049,3051,3069],{"className":3050,"ariaHidden":71},[70],[33,3052,3054,3057,3060,3063,3066],{"className":3053},[75],[33,3055],{"className":3056,"style":204},[79],[33,3058,2195],{"className":3059},[84,154],[33,3061],{"className":3062,"style":365},[364],[33,3064,276],{"className":3065},[369],[33,3067],{"className":3068,"style":365},[364],[33,3070,3072,3075],{"className":3071},[75],[33,3073],{"className":3074,"style":379},[79],[33,3076,925],{"className":3077},[84],[2253,3079,3080],{"align":2177},[33,3081,3083,3102],{"className":3082},[36],[33,3084,3086],{"className":3085},[40],[42,3087,3088],{"xmlns":44},[46,3089,3090,3099],{},[49,3091,3092],{},[52,3093,3094,3096],{},[59,3095,61],{},[59,3097,3098],{},"7",[63,3100,3101],{"encoding":65},"\\frac{4}{7}",[33,3103,3105],{"className":3104,"ariaHidden":71},[70],[33,3106,3108,3111],{"className":3107},[75],[33,3109],{"className":3110,"style":291},[79],[33,3112,3114,3117,3176],{"className":3113},[84],[33,3115],{"className":3116},[88,89],[33,3118,3120],{"className":3119},[52],[33,3121,3123,3168],{"className":3122},[96,97],[33,3124,3126,3165],{"className":3125},[101],[33,3127,3129,3143,3151],{"className":3128,"style":310},[105],[33,3130,3131,3134],{"style":109},[33,3132],{"className":3133,"style":114},[113],[33,3135,3137],{"className":3136},[118,119,120,121],[33,3138,3140],{"className":3139},[84,121],[33,3141,3098],{"className":3142},[84,121],[33,3144,3145,3148],{"style":130},[33,3146],{"className":3147,"style":114},[113],[33,3149],{"className":3150,"style":138},[137],[33,3152,3153,3156],{"style":141},[33,3154],{"className":3155,"style":114},[113],[33,3157,3159],{"className":3158},[118,119,120,121],[33,3160,3162],{"className":3161},[84,121],[33,3163,61],{"className":3164},[84,121],[33,3166,160],{"className":3167},[159],[33,3169,3171],{"className":3170},[101],[33,3172,3174],{"className":3173,"style":167},[105],[33,3175],{},[33,3177],{"className":3178},[173,89],[2253,3180,527],{"align":2177},[2253,3182,3183],{"align":2177},[33,3184,3186,3205],{"className":3185},[36],[33,3187,3189],{"className":3188},[40],[42,3190,3191],{"xmlns":44},[46,3192,3193,3202],{},[49,3194,3195],{},[52,3196,3197,3199],{},[59,3198,270],{},[59,3200,3201],{},"128",[63,3203,3204],{"encoding":65},"\\frac{1}{128}",[33,3206,3208],{"className":3207,"ariaHidden":71},[70],[33,3209,3211,3214],{"className":3210},[75],[33,3212],{"className":3213,"style":291},[79],[33,3215,3217,3220,3279],{"className":3216},[84],[33,3218],{"className":3219},[88,89],[33,3221,3223],{"className":3222},[52],[33,3224,3226,3271],{"className":3225},[96,97],[33,3227,3229,3268],{"className":3228},[101],[33,3230,3232,3246,3254],{"className":3231,"style":310},[105],[33,3233,3234,3237],{"style":109},[33,3235],{"className":3236,"style":114},[113],[33,3238,3240],{"className":3239},[118,119,120,121],[33,3241,3243],{"className":3242},[84,121],[33,3244,3201],{"className":3245},[84,121],[33,3247,3248,3251],{"style":130},[33,3249],{"className":3250,"style":114},[113],[33,3252],{"className":3253,"style":138},[137],[33,3255,3256,3259],{"style":141},[33,3257],{"className":3258,"style":114},[113],[33,3260,3262],{"className":3261},[118,119,120,121],[33,3263,3265],{"className":3264},[84,121],[33,3266,270],{"className":3267},[84,121],[33,3269,160],{"className":3270},[159],[33,3272,3274],{"className":3273},[101],[33,3275,3277],{"className":3276,"style":167},[105],[33,3278],{},[33,3280],{"className":3281},[173,89],[2172,3283,3284,3338,3439,3442],{},[2253,3285,3286,3287,1035],{"align":2177},"9 (",[33,3288,3290,3308],{"className":3289},[36],[33,3291,3293],{"className":3292},[40],[42,3294,3295],{"xmlns":44},[46,3296,3297,3305],{},[49,3298,3299,3301,3303],{},[55,3300,2195],{},[274,3302,276],{},[59,3304,61],{},[63,3306,3307],{"encoding":65},"n=4",[33,3309,3311,3329],{"className":3310,"ariaHidden":71},[70],[33,3312,3314,3317,3320,3323,3326],{"className":3313},[75],[33,3315],{"className":3316,"style":204},[79],[33,3318,2195],{"className":3319},[84,154],[33,3321],{"className":3322,"style":365},[364],[33,3324,276],{"className":3325},[369],[33,3327],{"className":3328,"style":365},[364],[33,3330,3332,3335],{"className":3331},[75],[33,3333],{"className":3334,"style":379},[79],[33,3336,61],{"className":3337},[84],[2253,3339,3340],{"align":2177},[33,3341,3343,3362],{"className":3342},[36],[33,3344,3346],{"className":3345},[40],[42,3347,3348],{"xmlns":44},[46,3349,3350,3359],{},[49,3351,3352],{},[52,3353,3354,3356],{},[59,3355,527],{},[59,3357,3358],{},"9",[63,3360,3361],{"encoding":65},"\\frac{5}{9}",[33,3363,3365],{"className":3364,"ariaHidden":71},[70],[33,3366,3368,3371],{"className":3367},[75],[33,3369],{"className":3370,"style":291},[79],[33,3372,3374,3377,3436],{"className":3373},[84],[33,3375],{"className":3376},[88,89],[33,3378,3380],{"className":3379},[52],[33,3381,3383,3428],{"className":3382},[96,97],[33,3384,3386,3425],{"className":3385},[101],[33,3387,3389,3403,3411],{"className":3388,"style":310},[105],[33,3390,3391,3394],{"style":109},[33,3392],{"className":3393,"style":114},[113],[33,3395,3397],{"className":3396},[118,119,120,121],[33,3398,3400],{"className":3399},[84,121],[33,3401,3358],{"className":3402},[84,121],[33,3404,3405,3408],{"style":130},[33,3406],{"className":3407,"style":114},[113],[33,3409],{"className":3410,"style":138},[137],[33,3412,3413,3416],{"style":141},[33,3414],{"className":3415,"style":114},[113],[33,3417,3419],{"className":3418},[118,119,120,121],[33,3420,3422],{"className":3421},[84,121],[33,3423,527],{"className":3424},[84,121],[33,3426,160],{"className":3427},[159],[33,3429,3431],{"className":3430},[101],[33,3432,3434],{"className":3433,"style":167},[105],[33,3435],{},[33,3437],{"className":3438},[173,89],[2253,3440,3441],{"align":2177},"14",[2253,3443,3444],{"align":2177},[33,3445,3447,3466],{"className":3446},[36],[33,3448,3450],{"className":3449},[40],[42,3451,3452],{"xmlns":44},[46,3453,3454,3463],{},[49,3455,3456],{},[52,3457,3458,3460],{},[59,3459,270],{},[59,3461,3462],{},"512",[63,3464,3465],{"encoding":65},"\\frac{1}{512}",[33,3467,3469],{"className":3468,"ariaHidden":71},[70],[33,3470,3472,3475],{"className":3471},[75],[33,3473],{"className":3474,"style":291},[79],[33,3476,3478,3481,3540],{"className":3477},[84],[33,3479],{"className":3480},[88,89],[33,3482,3484],{"className":3483},[52],[33,3485,3487,3532],{"className":3486},[96,97],[33,3488,3490,3529],{"className":3489},[101],[33,3491,3493,3507,3515],{"className":3492,"style":310},[105],[33,3494,3495,3498],{"style":109},[33,3496],{"className":3497,"style":114},[113],[33,3499,3501],{"className":3500},[118,119,120,121],[33,3502,3504],{"className":3503},[84,121],[33,3505,3462],{"className":3506},[84,121],[33,3508,3509,3512],{"style":130},[33,3510],{"className":3511,"style":114},[113],[33,3513],{"className":3514,"style":138},[137],[33,3516,3517,3520],{"style":141},[33,3518],{"className":3519,"style":114},[113],[33,3521,3523],{"className":3522},[118,119,120,121],[33,3524,3526],{"className":3525},[84,121],[33,3527,270],{"className":3528},[84,121],[33,3530,160],{"className":3531},[159],[33,3533,3535],{"className":3534},[101],[33,3536,3538],{"className":3537,"style":167},[105],[33,3539],{},[33,3541],{"className":3542},[173,89],[15,3544,3545],{},"Observa los patrones:",[3547,3548,3549,3608,3740],"ul",{},[230,3550,3551,3552,174],{},"La longitud siempre es impar: ",[33,3553,3555,3575],{"className":3554},[36],[33,3556,3558],{"className":3557},[40],[42,3559,3560],{"xmlns":44},[46,3561,3562,3572],{},[49,3563,3564,3566,3568,3570],{},[59,3565,402],{},[55,3567,2195],{},[274,3569,1314],{},[59,3571,270],{},[63,3573,3574],{"encoding":65},"2n + 1",[33,3576,3578,3599],{"className":3577,"ariaHidden":71},[70],[33,3579,3581,3584,3587,3590,3593,3596],{"className":3580},[75],[33,3582],{"className":3583,"style":1329},[79],[33,3585,402],{"className":3586},[84],[33,3588,2195],{"className":3589},[84,154],[33,3591],{"className":3592,"style":2221},[364],[33,3594,1314],{"className":3595},[2225],[33,3597],{"className":3598,"style":2221},[364],[33,3600,3602,3605],{"className":3601},[75],[33,3603],{"className":3604,"style":379},[79],[33,3606,270],{"className":3607},[84],[230,3609,3610,3611,174],{},"La proporción de caras es siempre ",[33,3612,3614,3646],{"className":3613},[36],[33,3615,3617],{"className":3616},[40],[42,3618,3619],{"xmlns":44},[46,3620,3621,3643],{},[49,3622,3623],{},[52,3624,3625,3633],{},[49,3626,3627,3629,3631],{},[55,3628,2195],{},[274,3630,1314],{},[59,3632,270],{},[49,3634,3635,3637,3639,3641],{},[59,3636,402],{},[55,3638,2195],{},[274,3640,1314],{},[59,3642,270],{},[63,3644,3645],{"encoding":65},"\\frac{n+1}{2n+1}",[33,3647,3649],{"className":3648,"ariaHidden":71},[70],[33,3650,3652,3656],{"className":3651},[75],[33,3653],{"className":3654,"style":3655},[79],"height:1.2484em;vertical-align:-0.4033em;",[33,3657,3659,3662,3737],{"className":3658},[84],[33,3660],{"className":3661},[88,89],[33,3663,3665],{"className":3664},[52],[33,3666,3668,3728],{"className":3667},[96,97],[33,3669,3671,3725],{"className":3670},[101],[33,3672,3674,3697,3705],{"className":3673,"style":310},[105],[33,3675,3676,3679],{"style":109},[33,3677],{"className":3678,"style":114},[113],[33,3680,3682],{"className":3681},[118,119,120,121],[33,3683,3685,3688,3691,3694],{"className":3684},[84,121],[33,3686,402],{"className":3687},[84,121],[33,3689,2195],{"className":3690},[84,154,121],[33,3692,1314],{"className":3693},[2225,121],[33,3695,270],{"className":3696},[84,121],[33,3698,3699,3702],{"style":130},[33,3700],{"className":3701,"style":114},[113],[33,3703],{"className":3704,"style":138},[137],[33,3706,3707,3710],{"style":141},[33,3708],{"className":3709,"style":114},[113],[33,3711,3713],{"className":3712},[118,119,120,121],[33,3714,3716,3719,3722],{"className":3715},[84,121],[33,3717,2195],{"className":3718},[84,154,121],[33,3720,1314],{"className":3721},[2225,121],[33,3723,270],{"className":3724},[84,121],[33,3726,160],{"className":3727},[159],[33,3729,3731],{"className":3730},[101],[33,3732,3735],{"className":3733,"style":3734},[105],"height:0.4033em;",[33,3736],{},[33,3738],{"className":3739},[173,89],[230,3741,3742,3743,3912,3913,3968],{},"La probabilidad de cada secuencia individual es ",[33,3744,3746,3782],{"className":3745},[36],[33,3747,3749],{"className":3748},[40],[42,3750,3751],{"xmlns":44},[46,3752,3753,3779],{},[49,3754,3755],{},[1020,3756,3757,3769],{},[49,3758,3759,3761,3767],{},[274,3760,1026],{"fence":71},[52,3762,3763,3765],{},[59,3764,270],{},[59,3766,402],{},[274,3768,1035],{"fence":71},[49,3770,3771,3773,3775,3777],{},[59,3772,402],{},[55,3774,2195],{},[274,3776,1314],{},[59,3778,270],{},[63,3780,3781],{"encoding":65},"\\left(\\frac{1}{2}\\right)^{2n+1}",[33,3783,3785],{"className":3784,"ariaHidden":71},[70],[33,3786,3788,3791],{"className":3787},[75],[33,3789],{"className":3790,"style":1059},[79],[33,3792,3794,3877],{"className":3793},[1063],[33,3795,3797,3803,3871],{"className":3796},[1063],[33,3798,3800],{"className":3799,"style":1071},[88,1070],[33,3801,1026],{"className":3802},[1075,1076],[33,3804,3806,3809,3868],{"className":3805},[84],[33,3807],{"className":3808},[88,89],[33,3810,3812],{"className":3811},[52],[33,3813,3815,3860],{"className":3814},[96,97],[33,3816,3818,3857],{"className":3817},[101],[33,3819,3821,3835,3843],{"className":3820,"style":310},[105],[33,3822,3823,3826],{"style":109},[33,3824],{"className":3825,"style":114},[113],[33,3827,3829],{"className":3828},[118,119,120,121],[33,3830,3832],{"className":3831},[84,121],[33,3833,402],{"className":3834},[84,121],[33,3836,3837,3840],{"style":130},[33,3838],{"className":3839,"style":114},[113],[33,3841],{"className":3842,"style":138},[137],[33,3844,3845,3848],{"style":141},[33,3846],{"className":3847,"style":114},[113],[33,3849,3851],{"className":3850},[118,119,120,121],[33,3852,3854],{"className":3853},[84,121],[33,3855,270],{"className":3856},[84,121],[33,3858,160],{"className":3859},[159],[33,3861,3863],{"className":3862},[101],[33,3864,3866],{"className":3865,"style":167},[105],[33,3867],{},[33,3869],{"className":3870},[173,89],[33,3872,3874],{"className":3873,"style":1071},[173,1070],[33,3875,1035],{"className":3876},[1075,1076],[33,3878,3880],{"className":3879},[1154],[33,3881,3883],{"className":3882},[96],[33,3884,3886],{"className":3885},[101],[33,3887,3889],{"className":3888,"style":1164},[105],[33,3890,3891,3894],{"style":1167},[33,3892],{"className":3893,"style":1171},[113],[33,3895,3897],{"className":3896},[118,119,120,121],[33,3898,3900,3903,3906,3909],{"className":3899},[84,121],[33,3901,402],{"className":3902},[84,121],[33,3904,2195],{"className":3905},[84,154,121],[33,3907,1314],{"className":3908},[2225,121],[33,3910,270],{"className":3911},[84,121]," — cada uno de los ",[33,3914,3916,3935],{"className":3915},[36],[33,3917,3919],{"className":3918},[40],[42,3920,3921],{"xmlns":44},[46,3922,3923,3933],{},[49,3924,3925,3927,3929,3931],{},[59,3926,402],{},[55,3928,2195],{},[274,3930,1314],{},[59,3932,270],{},[63,3934,2202],{"encoding":65},[33,3936,3938,3959],{"className":3937,"ariaHidden":71},[70],[33,3939,3941,3944,3947,3950,3953,3956],{"className":3940},[75],[33,3942],{"className":3943,"style":1329},[79],[33,3945,402],{"className":3946},[84],[33,3948,2195],{"className":3949},[84,154],[33,3951],{"className":3952,"style":2221},[364],[33,3954,1314],{"className":3955},[2225],[33,3957],{"className":3958,"style":2221},[364],[33,3960,3962,3965],{"className":3961},[75],[33,3963],{"className":3964,"style":379},[79],[33,3966,270],{"className":3967},[84]," lanzamientos tiene una probabilidad de un medio.",[15,3970,3971,3972],{},"Pero la columna que realmente importa es la de las secuencias válidas: ",[33,3973,3975,4010],{"className":3974},[36],[33,3976,3978],{"className":3977},[40],[42,3979,3980],{"xmlns":44},[46,3981,3982,4007],{},[49,3983,3984,3986,3988,3990,3992,3994,3996,3998,4000,4002,4004],{},[59,3985,270],{},[274,3987,523],{"separator":71},[59,3989,270],{},[274,3991,523],{"separator":71},[59,3993,402],{},[274,3995,523],{"separator":71},[59,3997,527],{},[274,3999,523],{"separator":71},[59,4001,3441],{},[274,4003,523],{"separator":71},[274,4005,4006],{},"…",[63,4008,4009],{"encoding":65},"1, 1, 2, 5, 14, \\ldots",[33,4011,4013],{"className":4012,"ariaHidden":71},[70],[33,4014,4016,4019,4022,4025,4028,4031,4034,4037,4040,4043,4046,4049,4052,4055,4058,4061,4064],{"className":4015},[75],[33,4017],{"className":4018,"style":511},[79],[33,4020,270],{"className":4021},[84],[33,4023,523],{"className":4024},[522],[33,4026],{"className":4027,"style":1450},[364],[33,4029,270],{"className":4030},[84],[33,4032,523],{"className":4033},[522],[33,4035],{"className":4036,"style":1450},[364],[33,4038,402],{"className":4039},[84],[33,4041,523],{"className":4042},[522],[33,4044],{"className":4045,"style":1450},[364],[33,4047,527],{"className":4048},[84],[33,4050,523],{"className":4051},[522],[33,4053],{"className":4054,"style":1450},[364],[33,4056,3441],{"className":4057},[84],[33,4059,523],{"className":4060},[522],[33,4062],{"className":4063,"style":1450},[364],[33,4065,4006],{"className":4066},[1063],[15,4068,4069],{},"¿Te suena?",[559,4071,4073],{"id":4072},"del-conteo-manual-a-los-números-de-catalan","Del conteo manual a los números de Catalan",[15,4075,4076,4077,4199,4200,4207,4208,4211],{},"Esos números — ",[33,4078,4080,4124],{"className":4079},[36],[33,4081,4083],{"className":4082},[40],[42,4084,4085],{"xmlns":44},[46,4086,4087,4121],{},[49,4088,4089,4091,4093,4095,4097,4099,4101,4103,4105,4107,4109,4112,4114,4117,4119],{},[59,4090,270],{},[274,4092,523],{"separator":71},[59,4094,270],{},[274,4096,523],{"separator":71},[59,4098,402],{},[274,4100,523],{"separator":71},[59,4102,527],{},[274,4104,523],{"separator":71},[59,4106,3441],{},[274,4108,523],{"separator":71},[59,4110,4111],{},"42",[274,4113,523],{"separator":71},[59,4115,4116],{},"132",[274,4118,523],{"separator":71},[274,4120,4006],{},[63,4122,4123],{"encoding":65},"1, 1, 2, 5, 14, 42, 132, \\ldots",[33,4125,4127],{"className":4126,"ariaHidden":71},[70],[33,4128,4130,4133,4136,4139,4142,4145,4148,4151,4154,4157,4160,4163,4166,4169,4172,4175,4178,4181,4184,4187,4190,4193,4196],{"className":4129},[75],[33,4131],{"className":4132,"style":511},[79],[33,4134,270],{"className":4135},[84],[33,4137,523],{"className":4138},[522],[33,4140],{"className":4141,"style":1450},[364],[33,4143,270],{"className":4144},[84],[33,4146,523],{"className":4147},[522],[33,4149],{"className":4150,"style":1450},[364],[33,4152,402],{"className":4153},[84],[33,4155,523],{"className":4156},[522],[33,4158],{"className":4159,"style":1450},[364],[33,4161,527],{"className":4162},[84],[33,4164,523],{"className":4165},[522],[33,4167],{"className":4168,"style":1450},[364],[33,4170,3441],{"className":4171},[84],[33,4173,523],{"className":4174},[522],[33,4176],{"className":4177,"style":1450},[364],[33,4179,4111],{"className":4180},[84],[33,4182,523],{"className":4183},[522],[33,4185],{"className":4186,"style":1450},[364],[33,4188,4116],{"className":4189},[84],[33,4191,523],{"className":4192},[522],[33,4194],{"className":4195,"style":1450},[364],[33,4197,4006],{"className":4198},[1063]," — tienen nombre propio. Son los ",[19,4201,4204],{"href":4202,"rel":4203},"https:\u002F\u002Fes.wikipedia.org\u002Fwiki\u002FN%C3%BAmeros_de_Catalan",[23],[237,4205,4206],{},"números de Catalan",", una de las secuencias más ubicuas de toda la combinatoria. (Y no, no son de origen ",[210,4209,4210],{},"Catalán",".)",[15,4213,4214,4215,4222,4223,4226,4227,4230,4231,4236,4237,4240],{},"Su historia es en sí misma una historia de conexiones inesperadas. El matemático mongol ",[19,4216,4219],{"href":4217,"rel":4218},"https:\u002F\u002Fes.wikipedia.org\u002Fwiki\u002FMinggatu",[23],[237,4220,4221],{},"Mingantu"," (c. 1692–1763), trabajando en China, fue la primera persona conocida en utilizarlos — alrededor de la década de 1730, en series trigonométricas incluidas en su libro ",[210,4224,4225],{},"Ge Yuan Mi Lu Jie Fa",". Dos décadas más tarde, en 1751, ",[237,4228,4229],{},"Leonhard Euler"," ",[19,4232,4235],{"href":4233,"rel":4234},"https:\u002F\u002Fes.wikipedia.org\u002Fwiki\u002FN%C3%BAmeros_de_Catalan#Historia",[23],"descubrió la misma secuencia de forma independiente"," en una carta a Goldbach, esta vez contando las formas de dividir un polígono en triángulos. Y casi un siglo después de Mingantu, el matemático franco-belga ",[237,4238,4239],{},"Eugène Charles Catalan"," (1814–1894) los encontró de nuevo en 1838, al contar las formas de colocar paréntesis en un producto de factores. La secuencia lleva su nombre — acuñado por John Riordan en el siglo XX — a pesar de que los números habían estado apareciendo, bajo diferentes formas, durante más de cien años antes que él.",[15,4242,4243],{},"La misma secuencia, descubierta de forma independiente en tres partes del mundo y a lo largo de tres siglos, cada vez en un contexto completamente distinto. Si ese patrón te suena familiar... sigue leyendo.",[15,4245,4246,4247,4275],{},"El ",[33,4248,4250,4263],{"className":4249},[36],[33,4251,4253],{"className":4252},[40],[42,4254,4255],{"xmlns":44},[46,4256,4257,4261],{},[49,4258,4259],{},[55,4260,2195],{},[63,4262,2195],{"encoding":65},[33,4264,4266],{"className":4265,"ariaHidden":71},[70],[33,4267,4269,4272],{"className":4268},[75],[33,4270],{"className":4271,"style":204},[79],[33,4273,2195],{"className":4274},[84,154],"-ésimo número de Catalan se define como:",[15,4277,4279],{"className":4278},[590],[33,4280,4282,4333],{"className":4281},[36],[33,4283,4285],{"className":4284},[40],[42,4286,4287],{"xmlns":44},[46,4288,4289,4330],{},[49,4290,4291,4299,4301,4313],{},[4292,4293,4294,4297],"msub",{},[55,4295,4296],{},"C",[55,4298,2195],{},[274,4300,276],{},[52,4302,4303,4305],{},[59,4304,270],{},[49,4306,4307,4309,4311],{},[55,4308,2195],{},[274,4310,1314],{},[59,4312,270],{},[49,4314,4315,4317,4328],{},[274,4316,1026],{"fence":71},[52,4318,4320,4326],{"linethickness":4319},"0px",[49,4321,4322,4324],{},[59,4323,402],{},[55,4325,2195],{},[55,4327,2195],{},[274,4329,1035],{"fence":71},[63,4331,4332],{"encoding":65},"C_n = \\frac{1}{n+1}\\binom{2n}{n}",[33,4334,4336,4396],{"className":4335,"ariaHidden":71},[70],[33,4337,4339,4343,4387,4390,4393],{"className":4338},[75],[33,4340],{"className":4341,"style":4342},[79],"height:0.8333em;vertical-align:-0.15em;",[33,4344,4346,4350],{"className":4345},[84],[33,4347,4296],{"className":4348,"style":4349},[84,154],"margin-right:0.0715em;",[33,4351,4353],{"className":4352},[1154],[33,4354,4356,4378],{"className":4355},[96,97],[33,4357,4359,4375],{"className":4358},[101],[33,4360,4363],{"className":4361,"style":4362},[105],"height:0.1514em;",[33,4364,4366,4369],{"style":4365},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[33,4367],{"className":4368,"style":1171},[113],[33,4370,4372],{"className":4371},[118,119,120,121],[33,4373,2195],{"className":4374},[84,154,121],[33,4376,160],{"className":4377},[159],[33,4379,4381],{"className":4380},[101],[33,4382,4385],{"className":4383,"style":4384},[105],"height:0.15em;",[33,4386],{},[33,4388],{"className":4389,"style":365},[364],[33,4391,276],{"className":4392},[369],[33,4394],{"className":4395,"style":365},[364],[33,4397,4399,4403,4477],{"className":4398},[75],[33,4400],{"className":4401,"style":4402},[79],"height:1.2984em;vertical-align:-0.4033em;",[33,4404,4406,4409,4474],{"className":4405},[84],[33,4407],{"className":4408},[88,89],[33,4410,4412],{"className":4411},[52],[33,4413,4415,4466],{"className":4414},[96,97],[33,4416,4418,4463],{"className":4417},[101],[33,4419,4421,4441,4449],{"className":4420,"style":310},[105],[33,4422,4423,4426],{"style":109},[33,4424],{"className":4425,"style":114},[113],[33,4427,4429],{"className":4428},[118,119,120,121],[33,4430,4432,4435,4438],{"className":4431},[84,121],[33,4433,2195],{"className":4434},[84,154,121],[33,4436,1314],{"className":4437},[2225,121],[33,4439,270],{"className":4440},[84,121],[33,4442,4443,4446],{"style":130},[33,4444],{"className":4445,"style":114},[113],[33,4447],{"className":4448,"style":138},[137],[33,4450,4451,4454],{"style":141},[33,4452],{"className":4453,"style":114},[113],[33,4455,4457],{"className":4456},[118,119,120,121],[33,4458,4460],{"className":4459},[84,121],[33,4461,270],{"className":4462},[84,121],[33,4464,160],{"className":4465},[159],[33,4467,4469],{"className":4468},[101],[33,4470,4472],{"className":4471,"style":3734},[105],[33,4473],{},[33,4475],{"className":4476},[173,89],[33,4478,4480,4486,4543],{"className":4479},[84],[33,4481,4483],{"className":4482,"style":1071},[88,1070],[33,4484,1026],{"className":4485},[1075,1076],[33,4487,4489],{"className":4488},[52],[33,4490,4492,4535],{"className":4491},[96,97],[33,4493,4495,4532],{"className":4494},[101],[33,4496,4499,4514],{"className":4497,"style":4498},[105],"height:0.8951em;",[33,4500,4502,4505],{"style":4501},"top:-2.355em;",[33,4503],{"className":4504,"style":1171},[113],[33,4506,4508],{"className":4507},[118,119,120,121],[33,4509,4511],{"className":4510},[84,121],[33,4512,2195],{"className":4513},[84,154,121],[33,4515,4517,4520],{"style":4516},"top:-3.144em;",[33,4518],{"className":4519,"style":1171},[113],[33,4521,4523],{"className":4522},[118,119,120,121],[33,4524,4526,4529],{"className":4525},[84,121],[33,4527,402],{"className":4528},[84,121],[33,4530,2195],{"className":4531},[84,154,121],[33,4533,160],{"className":4534},[159],[33,4536,4538],{"className":4537},[101],[33,4539,4541],{"className":4540,"style":167},[105],[33,4542],{},[33,4544,4546],{"className":4545,"style":1071},[173,1070],[33,4547,1035],{"className":4548},[1075,1076],[15,4550,4551,4552,4660,4661,4695,4696,4724,4725,4728,4729,4757,4758,4791,4792,4820,4821,4854,4855,174],{},"donde ",[33,4553,4555,4581],{"className":4554},[36],[33,4556,4558],{"className":4557},[40],[42,4559,4560],{"xmlns":44},[46,4561,4562,4578],{},[49,4563,4564,4566,4576],{},[274,4565,1026],{"fence":71},[52,4567,4568,4574],{"linethickness":4319},[49,4569,4570,4572],{},[59,4571,402],{},[55,4573,2195],{},[55,4575,2195],{},[274,4577,1035],{"fence":71},[63,4579,4580],{"encoding":65},"\\binom{2n}{n}",[33,4582,4584],{"className":4583,"ariaHidden":71},[70],[33,4585,4587,4591],{"className":4586},[75],[33,4588],{"className":4589,"style":4590},[79],"height:1.2451em;vertical-align:-0.35em;",[33,4592,4594,4600,4654],{"className":4593},[84],[33,4595,4597],{"className":4596,"style":1071},[88,1070],[33,4598,1026],{"className":4599},[1075,1076],[33,4601,4603],{"className":4602},[52],[33,4604,4606,4646],{"className":4605},[96,97],[33,4607,4609,4643],{"className":4608},[101],[33,4610,4612,4626],{"className":4611,"style":4498},[105],[33,4613,4614,4617],{"style":4501},[33,4615],{"className":4616,"style":1171},[113],[33,4618,4620],{"className":4619},[118,119,120,121],[33,4621,4623],{"className":4622},[84,121],[33,4624,2195],{"className":4625},[84,154,121],[33,4627,4628,4631],{"style":4516},[33,4629],{"className":4630,"style":1171},[113],[33,4632,4634],{"className":4633},[118,119,120,121],[33,4635,4637,4640],{"className":4636},[84,121],[33,4638,402],{"className":4639},[84,121],[33,4641,2195],{"className":4642},[84,154,121],[33,4644,160],{"className":4645},[159],[33,4647,4649],{"className":4648},[101],[33,4650,4652],{"className":4651,"style":167},[105],[33,4653],{},[33,4655,4657],{"className":4656,"style":1071},[173,1070],[33,4658,1035],{"className":4659},[1075,1076]," — se lee «",[33,4662,4664,4680],{"className":4663},[36],[33,4665,4667],{"className":4666},[40],[42,4668,4669],{"xmlns":44},[46,4670,4671,4677],{},[49,4672,4673,4675],{},[59,4674,402],{},[55,4676,2195],{},[63,4678,4679],{"encoding":65},"2n",[33,4681,4683],{"className":4682,"ariaHidden":71},[70],[33,4684,4686,4689,4692],{"className":4685},[75],[33,4687],{"className":4688,"style":379},[79],[33,4690,402],{"className":4691},[84],[33,4693,2195],{"className":4694},[84,154]," sobre ",[33,4697,4699,4712],{"className":4698},[36],[33,4700,4702],{"className":4701},[40],[42,4703,4704],{"xmlns":44},[46,4705,4706,4710],{},[49,4707,4708],{},[55,4709,2195],{},[63,4711,2195],{"encoding":65},[33,4713,4715],{"className":4714,"ariaHidden":71},[70],[33,4716,4718,4721],{"className":4717},[75],[33,4719],{"className":4720,"style":204},[79],[33,4722,2195],{"className":4723},[84,154],"» — es el ",[237,4726,4727],{},"coeficiente binomial",": el número de formas de elegir ",[33,4730,4732,4745],{"className":4731},[36],[33,4733,4735],{"className":4734},[40],[42,4736,4737],{"xmlns":44},[46,4738,4739,4743],{},[49,4740,4741],{},[55,4742,2195],{},[63,4744,2195],{"encoding":65},[33,4746,4748],{"className":4747,"ariaHidden":71},[70],[33,4749,4751,4754],{"className":4750},[75],[33,4752],{"className":4753,"style":204},[79],[33,4755,2195],{"className":4756},[84,154]," elementos de un conjunto de ",[33,4759,4761,4776],{"className":4760},[36],[33,4762,4764],{"className":4763},[40],[42,4765,4766],{"xmlns":44},[46,4767,4768,4774],{},[49,4769,4770,4772],{},[59,4771,402],{},[55,4773,2195],{},[63,4775,4679],{"encoding":65},[33,4777,4779],{"className":4778,"ariaHidden":71},[70],[33,4780,4782,4785,4788],{"className":4781},[75],[33,4783],{"className":4784,"style":379},[79],[33,4786,402],{"className":4787},[84],[33,4789,2195],{"className":4790},[84,154],". En la práctica, responde a preguntas como «¿de cuántas formas puedo distribuir ",[33,4793,4795,4808],{"className":4794},[36],[33,4796,4798],{"className":4797},[40],[42,4799,4800],{"xmlns":44},[46,4801,4802,4806],{},[49,4803,4804],{},[55,4805,2195],{},[63,4807,2195],{"encoding":65},[33,4809,4811],{"className":4810,"ariaHidden":71},[70],[33,4812,4814,4817],{"className":4813},[75],[33,4815],{"className":4816,"style":204},[79],[33,4818,2195],{"className":4819},[84,154]," caras en ",[33,4822,4824,4839],{"className":4823},[36],[33,4825,4827],{"className":4826},[40],[42,4828,4829],{"xmlns":44},[46,4830,4831,4837],{},[49,4832,4833,4835],{},[59,4834,402],{},[55,4836,2195],{},[63,4838,4679],{"encoding":65},[33,4840,4842],{"className":4841,"ariaHidden":71},[70],[33,4843,4845,4848,4851],{"className":4844},[75],[33,4846],{"className":4847,"style":379},[79],[33,4849,402],{"className":4850},[84],[33,4852,2195],{"className":4853},[84,154]," lanzamientos?» Exploramos esta notación en detalle en el ",[19,4856,4858],{"href":219,"rel":4857},[23],"artículo sobre el triángulo de Tartaglia",[15,4860,4861],{},"Comprobemos con nuestros datos:",[15,4863,4865],{"className":4864},[590],[33,4866,4868,4943],{"className":4867},[36],[33,4869,4871],{"className":4870},[40],[42,4872,4873],{"xmlns":44},[46,4874,4875,4940],{},[49,4876,4877,4883,4885,4891,4903,4905,4907,4910,4916,4918,4924,4936,4938],{},[4292,4878,4879,4881],{},[55,4880,4296],{},[59,4882,515],{},[274,4884,276],{},[52,4886,4887,4889],{},[59,4888,270],{},[59,4890,270],{},[49,4892,4893,4895,4901],{},[274,4894,1026],{"fence":71},[52,4896,4897,4899],{"linethickness":4319},[59,4898,515],{},[59,4900,515],{},[274,4902,1035],{"fence":71},[274,4904,276],{},[59,4906,270],{},[364,4908],{"width":4909},"2em",[4292,4911,4912,4914],{},[55,4913,4296],{},[59,4915,270],{},[274,4917,276],{},[52,4919,4920,4922],{},[59,4921,270],{},[59,4923,402],{},[49,4925,4926,4928,4934],{},[274,4927,1026],{"fence":71},[52,4929,4930,4932],{"linethickness":4319},[59,4931,402],{},[59,4933,270],{},[274,4935,1035],{"fence":71},[274,4937,276],{},[59,4939,270],{},[63,4941,4942],{"encoding":65},"C_0 = \\frac{1}{1}\\binom{0}{0} = 1 \\qquad C_1 = \\frac{1}{2}\\binom{2}{1} = 1",[33,4944,4946,5002,5151,5213,5362],{"className":4945,"ariaHidden":71},[70],[33,4947,4949,4952,4993,4996,4999],{"className":4948},[75],[33,4950],{"className":4951,"style":4342},[79],[33,4953,4955,4958],{"className":4954},[84],[33,4956,4296],{"className":4957,"style":4349},[84,154],[33,4959,4961],{"className":4960},[1154],[33,4962,4964,4985],{"className":4963},[96,97],[33,4965,4967,4982],{"className":4966},[101],[33,4968,4971],{"className":4969,"style":4970},[105],"height:0.3011em;",[33,4972,4973,4976],{"style":4365},[33,4974],{"className":4975,"style":1171},[113],[33,4977,4979],{"className":4978},[118,119,120,121],[33,4980,515],{"className":4981},[84,121],[33,4983,160],{"className":4984},[159],[33,4986,4988],{"className":4987},[101],[33,4989,4991],{"className":4990,"style":4384},[105],[33,4992],{},[33,4994],{"className":4995,"style":365},[364],[33,4997,276],{"className":4998},[369],[33,5000],{"className":5001,"style":365},[364],[33,5003,5005,5008,5076,5142,5145,5148],{"className":5004},[75],[33,5006],{"className":5007,"style":4590},[79],[33,5009,5011,5014,5073],{"className":5010},[84],[33,5012],{"className":5013},[88,89],[33,5015,5017],{"className":5016},[52],[33,5018,5020,5065],{"className":5019},[96,97],[33,5021,5023,5062],{"className":5022},[101],[33,5024,5026,5040,5048],{"className":5025,"style":310},[105],[33,5027,5028,5031],{"style":109},[33,5029],{"className":5030,"style":114},[113],[33,5032,5034],{"className":5033},[118,119,120,121],[33,5035,5037],{"className":5036},[84,121],[33,5038,270],{"className":5039},[84,121],[33,5041,5042,5045],{"style":130},[33,5043],{"className":5044,"style":114},[113],[33,5046],{"className":5047,"style":138},[137],[33,5049,5050,5053],{"style":141},[33,5051],{"className":5052,"style":114},[113],[33,5054,5056],{"className":5055},[118,119,120,121],[33,5057,5059],{"className":5058},[84,121],[33,5060,270],{"className":5061},[84,121],[33,5063,160],{"className":5064},[159],[33,5066,5068],{"className":5067},[101],[33,5069,5071],{"className":5070,"style":167},[105],[33,5072],{},[33,5074],{"className":5075},[173,89],[33,5077,5079,5085,5136],{"className":5078},[84],[33,5080,5082],{"className":5081,"style":1071},[88,1070],[33,5083,1026],{"className":5084},[1075,1076],[33,5086,5088],{"className":5087},[52],[33,5089,5091,5128],{"className":5090},[96,97],[33,5092,5094,5125],{"className":5093},[101],[33,5095,5097,5111],{"className":5096,"style":4498},[105],[33,5098,5099,5102],{"style":4501},[33,5100],{"className":5101,"style":1171},[113],[33,5103,5105],{"className":5104},[118,119,120,121],[33,5106,5108],{"className":5107},[84,121],[33,5109,515],{"className":5110},[84,121],[33,5112,5113,5116],{"style":4516},[33,5114],{"className":5115,"style":1171},[113],[33,5117,5119],{"className":5118},[118,119,120,121],[33,5120,5122],{"className":5121},[84,121],[33,5123,515],{"className":5124},[84,121],[33,5126,160],{"className":5127},[159],[33,5129,5131],{"className":5130},[101],[33,5132,5134],{"className":5133,"style":167},[105],[33,5135],{},[33,5137,5139],{"className":5138,"style":1071},[173,1070],[33,5140,1035],{"className":5141},[1075,1076],[33,5143],{"className":5144,"style":365},[364],[33,5146,276],{"className":5147},[369],[33,5149],{"className":5150,"style":365},[364],[33,5152,5154,5157,5160,5164,5204,5207,5210],{"className":5153},[75],[33,5155],{"className":5156,"style":4342},[79],[33,5158,270],{"className":5159},[84],[33,5161],{"className":5162,"style":5163},[364],"margin-right:2em;",[33,5165,5167,5170],{"className":5166},[84],[33,5168,4296],{"className":5169,"style":4349},[84,154],[33,5171,5173],{"className":5172},[1154],[33,5174,5176,5196],{"className":5175},[96,97],[33,5177,5179,5193],{"className":5178},[101],[33,5180,5182],{"className":5181,"style":4970},[105],[33,5183,5184,5187],{"style":4365},[33,5185],{"className":5186,"style":1171},[113],[33,5188,5190],{"className":5189},[118,119,120,121],[33,5191,270],{"className":5192},[84,121],[33,5194,160],{"className":5195},[159],[33,5197,5199],{"className":5198},[101],[33,5200,5202],{"className":5201,"style":4384},[105],[33,5203],{},[33,5205],{"className":5206,"style":365},[364],[33,5208,276],{"className":5209},[369],[33,5211],{"className":5212,"style":365},[364],[33,5214,5216,5219,5287,5353,5356,5359],{"className":5215},[75],[33,5217],{"className":5218,"style":4590},[79],[33,5220,5222,5225,5284],{"className":5221},[84],[33,5223],{"className":5224},[88,89],[33,5226,5228],{"className":5227},[52],[33,5229,5231,5276],{"className":5230},[96,97],[33,5232,5234,5273],{"className":5233},[101],[33,5235,5237,5251,5259],{"className":5236,"style":310},[105],[33,5238,5239,5242],{"style":109},[33,5240],{"className":5241,"style":114},[113],[33,5243,5245],{"className":5244},[118,119,120,121],[33,5246,5248],{"className":5247},[84,121],[33,5249,402],{"className":5250},[84,121],[33,5252,5253,5256],{"style":130},[33,5254],{"className":5255,"style":114},[113],[33,5257],{"className":5258,"style":138},[137],[33,5260,5261,5264],{"style":141},[33,5262],{"className":5263,"style":114},[113],[33,5265,5267],{"className":5266},[118,119,120,121],[33,5268,5270],{"className":5269},[84,121],[33,5271,270],{"className":5272},[84,121],[33,5274,160],{"className":5275},[159],[33,5277,5279],{"className":5278},[101],[33,5280,5282],{"className":5281,"style":167},[105],[33,5283],{},[33,5285],{"className":5286},[173,89],[33,5288,5290,5296,5347],{"className":5289},[84],[33,5291,5293],{"className":5292,"style":1071},[88,1070],[33,5294,1026],{"className":5295},[1075,1076],[33,5297,5299],{"className":5298},[52],[33,5300,5302,5339],{"className":5301},[96,97],[33,5303,5305,5336],{"className":5304},[101],[33,5306,5308,5322],{"className":5307,"style":4498},[105],[33,5309,5310,5313],{"style":4501},[33,5311],{"className":5312,"style":1171},[113],[33,5314,5316],{"className":5315},[118,119,120,121],[33,5317,5319],{"className":5318},[84,121],[33,5320,270],{"className":5321},[84,121],[33,5323,5324,5327],{"style":4516},[33,5325],{"className":5326,"style":1171},[113],[33,5328,5330],{"className":5329},[118,119,120,121],[33,5331,5333],{"className":5332},[84,121],[33,5334,402],{"className":5335},[84,121],[33,5337,160],{"className":5338},[159],[33,5340,5342],{"className":5341},[101],[33,5343,5345],{"className":5344,"style":167},[105],[33,5346],{},[33,5348,5350],{"className":5349,"style":1071},[173,1070],[33,5351,1035],{"className":5352},[1075,1076],[33,5354],{"className":5355,"style":365},[364],[33,5357,276],{"className":5358},[369],[33,5360],{"className":5361,"style":365},[364],[33,5363,5365,5368],{"className":5364},[75],[33,5366],{"className":5367,"style":379},[79],[33,5369,270],{"className":5370},[84],[15,5372,5374],{"className":5373},[590],[33,5375,5377,5452],{"className":5376},[36],[33,5378,5380],{"className":5379},[40],[42,5381,5382],{"xmlns":44},[46,5383,5384,5449],{},[49,5385,5386,5392,5394,5400,5412,5414,5416,5418,5424,5426,5432,5445,5447],{},[4292,5387,5388,5390],{},[55,5389,4296],{},[59,5391,402],{},[274,5393,276],{},[52,5395,5396,5398],{},[59,5397,270],{},[59,5399,925],{},[49,5401,5402,5404,5410],{},[274,5403,1026],{"fence":71},[52,5405,5406,5408],{"linethickness":4319},[59,5407,61],{},[59,5409,402],{},[274,5411,1035],{"fence":71},[274,5413,276],{},[59,5415,402],{},[364,5417],{"width":4909},[4292,5419,5420,5422],{},[55,5421,4296],{},[59,5423,925],{},[274,5425,276],{},[52,5427,5428,5430],{},[59,5429,270],{},[59,5431,61],{},[49,5433,5434,5436,5443],{},[274,5435,1026],{"fence":71},[52,5437,5438,5441],{"linethickness":4319},[59,5439,5440],{},"6",[59,5442,925],{},[274,5444,1035],{"fence":71},[274,5446,276],{},[59,5448,527],{},[63,5450,5451],{"encoding":65},"C_2 = \\frac{1}{3}\\binom{4}{2} = 2 \\qquad C_3 = \\frac{1}{4}\\binom{6}{3} = 5 ",[33,5453,5455,5510,5659,5720,5869],{"className":5454,"ariaHidden":71},[70],[33,5456,5458,5461,5501,5504,5507],{"className":5457},[75],[33,5459],{"className":5460,"style":4342},[79],[33,5462,5464,5467],{"className":5463},[84],[33,5465,4296],{"className":5466,"style":4349},[84,154],[33,5468,5470],{"className":5469},[1154],[33,5471,5473,5493],{"className":5472},[96,97],[33,5474,5476,5490],{"className":5475},[101],[33,5477,5479],{"className":5478,"style":4970},[105],[33,5480,5481,5484],{"style":4365},[33,5482],{"className":5483,"style":1171},[113],[33,5485,5487],{"className":5486},[118,119,120,121],[33,5488,402],{"className":5489},[84,121],[33,5491,160],{"className":5492},[159],[33,5494,5496],{"className":5495},[101],[33,5497,5499],{"className":5498,"style":4384},[105],[33,5500],{},[33,5502],{"className":5503,"style":365},[364],[33,5505,276],{"className":5506},[369],[33,5508],{"className":5509,"style":365},[364],[33,5511,5513,5516,5584,5650,5653,5656],{"className":5512},[75],[33,5514],{"className":5515,"style":4590},[79],[33,5517,5519,5522,5581],{"className":5518},[84],[33,5520],{"className":5521},[88,89],[33,5523,5525],{"className":5524},[52],[33,5526,5528,5573],{"className":5527},[96,97],[33,5529,5531,5570],{"className":5530},[101],[33,5532,5534,5548,5556],{"className":5533,"style":310},[105],[33,5535,5536,5539],{"style":109},[33,5537],{"className":5538,"style":114},[113],[33,5540,5542],{"className":5541},[118,119,120,121],[33,5543,5545],{"className":5544},[84,121],[33,5546,925],{"className":5547},[84,121],[33,5549,5550,5553],{"style":130},[33,5551],{"className":5552,"style":114},[113],[33,5554],{"className":5555,"style":138},[137],[33,5557,5558,5561],{"style":141},[33,5559],{"className":5560,"style":114},[113],[33,5562,5564],{"className":5563},[118,119,120,121],[33,5565,5567],{"className":5566},[84,121],[33,5568,270],{"className":5569},[84,121],[33,5571,160],{"className":5572},[159],[33,5574,5576],{"className":5575},[101],[33,5577,5579],{"className":5578,"style":167},[105],[33,5580],{},[33,5582],{"className":5583},[173,89],[33,5585,5587,5593,5644],{"className":5586},[84],[33,5588,5590],{"className":5589,"style":1071},[88,1070],[33,5591,1026],{"className":5592},[1075,1076],[33,5594,5596],{"className":5595},[52],[33,5597,5599,5636],{"className":5598},[96,97],[33,5600,5602,5633],{"className":5601},[101],[33,5603,5605,5619],{"className":5604,"style":4498},[105],[33,5606,5607,5610],{"style":4501},[33,5608],{"className":5609,"style":1171},[113],[33,5611,5613],{"className":5612},[118,119,120,121],[33,5614,5616],{"className":5615},[84,121],[33,5617,402],{"className":5618},[84,121],[33,5620,5621,5624],{"style":4516},[33,5622],{"className":5623,"style":1171},[113],[33,5625,5627],{"className":5626},[118,119,120,121],[33,5628,5630],{"className":5629},[84,121],[33,5631,61],{"className":5632},[84,121],[33,5634,160],{"className":5635},[159],[33,5637,5639],{"className":5638},[101],[33,5640,5642],{"className":5641,"style":167},[105],[33,5643],{},[33,5645,5647],{"className":5646,"style":1071},[173,1070],[33,5648,1035],{"className":5649},[1075,1076],[33,5651],{"className":5652,"style":365},[364],[33,5654,276],{"className":5655},[369],[33,5657],{"className":5658,"style":365},[364],[33,5660,5662,5665,5668,5671,5711,5714,5717],{"className":5661},[75],[33,5663],{"className":5664,"style":4342},[79],[33,5666,402],{"className":5667},[84],[33,5669],{"className":5670,"style":5163},[364],[33,5672,5674,5677],{"className":5673},[84],[33,5675,4296],{"className":5676,"style":4349},[84,154],[33,5678,5680],{"className":5679},[1154],[33,5681,5683,5703],{"className":5682},[96,97],[33,5684,5686,5700],{"className":5685},[101],[33,5687,5689],{"className":5688,"style":4970},[105],[33,5690,5691,5694],{"style":4365},[33,5692],{"className":5693,"style":1171},[113],[33,5695,5697],{"className":5696},[118,119,120,121],[33,5698,925],{"className":5699},[84,121],[33,5701,160],{"className":5702},[159],[33,5704,5706],{"className":5705},[101],[33,5707,5709],{"className":5708,"style":4384},[105],[33,5710],{},[33,5712],{"className":5713,"style":365},[364],[33,5715,276],{"className":5716},[369],[33,5718],{"className":5719,"style":365},[364],[33,5721,5723,5726,5794,5860,5863,5866],{"className":5722},[75],[33,5724],{"className":5725,"style":4590},[79],[33,5727,5729,5732,5791],{"className":5728},[84],[33,5730],{"className":5731},[88,89],[33,5733,5735],{"className":5734},[52],[33,5736,5738,5783],{"className":5737},[96,97],[33,5739,5741,5780],{"className":5740},[101],[33,5742,5744,5758,5766],{"className":5743,"style":310},[105],[33,5745,5746,5749],{"style":109},[33,5747],{"className":5748,"style":114},[113],[33,5750,5752],{"className":5751},[118,119,120,121],[33,5753,5755],{"className":5754},[84,121],[33,5756,61],{"className":5757},[84,121],[33,5759,5760,5763],{"style":130},[33,5761],{"className":5762,"style":114},[113],[33,5764],{"className":5765,"style":138},[137],[33,5767,5768,5771],{"style":141},[33,5769],{"className":5770,"style":114},[113],[33,5772,5774],{"className":5773},[118,119,120,121],[33,5775,5777],{"className":5776},[84,121],[33,5778,270],{"className":5779},[84,121],[33,5781,160],{"className":5782},[159],[33,5784,5786],{"className":5785},[101],[33,5787,5789],{"className":5788,"style":167},[105],[33,5790],{},[33,5792],{"className":5793},[173,89],[33,5795,5797,5803,5854],{"className":5796},[84],[33,5798,5800],{"className":5799,"style":1071},[88,1070],[33,5801,1026],{"className":5802},[1075,1076],[33,5804,5806],{"className":5805},[52],[33,5807,5809,5846],{"className":5808},[96,97],[33,5810,5812,5843],{"className":5811},[101],[33,5813,5815,5829],{"className":5814,"style":4498},[105],[33,5816,5817,5820],{"style":4501},[33,5818],{"className":5819,"style":1171},[113],[33,5821,5823],{"className":5822},[118,119,120,121],[33,5824,5826],{"className":5825},[84,121],[33,5827,925],{"className":5828},[84,121],[33,5830,5831,5834],{"style":4516},[33,5832],{"className":5833,"style":1171},[113],[33,5835,5837],{"className":5836},[118,119,120,121],[33,5838,5840],{"className":5839},[84,121],[33,5841,5440],{"className":5842},[84,121],[33,5844,160],{"className":5845},[159],[33,5847,5849],{"className":5848},[101],[33,5850,5852],{"className":5851,"style":167},[105],[33,5853],{},[33,5855,5857],{"className":5856,"style":1071},[173,1070],[33,5858,1035],{"className":5859},[1075,1076],[33,5861],{"className":5862,"style":365},[364],[33,5864,276],{"className":5865},[369],[33,5867],{"className":5868,"style":365},[364],[33,5870,5872,5875],{"className":5871},[75],[33,5873],{"className":5874,"style":379},[79],[33,5876,527],{"className":5877},[84],[15,5879,5881],{"className":5880},[590],[33,5882,5884,5926],{"className":5883},[36],[33,5885,5887],{"className":5886},[40],[42,5888,5889],{"xmlns":44},[46,5890,5891,5923],{},[49,5892,5893,5899,5901,5907,5919,5921],{},[4292,5894,5895,5897],{},[55,5896,4296],{},[59,5898,61],{},[274,5900,276],{},[52,5902,5903,5905],{},[59,5904,270],{},[59,5906,527],{},[49,5908,5909,5911,5917],{},[274,5910,1026],{"fence":71},[52,5912,5913,5915],{"linethickness":4319},[59,5914,1046],{},[59,5916,61],{},[274,5918,1035],{"fence":71},[274,5920,276],{},[59,5922,3441],{},[63,5924,5925],{"encoding":65},"C_4 = \\frac{1}{5}\\binom{8}{4} = 14",[33,5927,5929,5984,6133],{"className":5928,"ariaHidden":71},[70],[33,5930,5932,5935,5975,5978,5981],{"className":5931},[75],[33,5933],{"className":5934,"style":4342},[79],[33,5936,5938,5941],{"className":5937},[84],[33,5939,4296],{"className":5940,"style":4349},[84,154],[33,5942,5944],{"className":5943},[1154],[33,5945,5947,5967],{"className":5946},[96,97],[33,5948,5950,5964],{"className":5949},[101],[33,5951,5953],{"className":5952,"style":4970},[105],[33,5954,5955,5958],{"style":4365},[33,5956],{"className":5957,"style":1171},[113],[33,5959,5961],{"className":5960},[118,119,120,121],[33,5962,61],{"className":5963},[84,121],[33,5965,160],{"className":5966},[159],[33,5968,5970],{"className":5969},[101],[33,5971,5973],{"className":5972,"style":4384},[105],[33,5974],{},[33,5976],{"className":5977,"style":365},[364],[33,5979,276],{"className":5980},[369],[33,5982],{"className":5983,"style":365},[364],[33,5985,5987,5990,6058,6124,6127,6130],{"className":5986},[75],[33,5988],{"className":5989,"style":4590},[79],[33,5991,5993,5996,6055],{"className":5992},[84],[33,5994],{"className":5995},[88,89],[33,5997,5999],{"className":5998},[52],[33,6000,6002,6047],{"className":6001},[96,97],[33,6003,6005,6044],{"className":6004},[101],[33,6006,6008,6022,6030],{"className":6007,"style":310},[105],[33,6009,6010,6013],{"style":109},[33,6011],{"className":6012,"style":114},[113],[33,6014,6016],{"className":6015},[118,119,120,121],[33,6017,6019],{"className":6018},[84,121],[33,6020,527],{"className":6021},[84,121],[33,6023,6024,6027],{"style":130},[33,6025],{"className":6026,"style":114},[113],[33,6028],{"className":6029,"style":138},[137],[33,6031,6032,6035],{"style":141},[33,6033],{"className":6034,"style":114},[113],[33,6036,6038],{"className":6037},[118,119,120,121],[33,6039,6041],{"className":6040},[84,121],[33,6042,270],{"className":6043},[84,121],[33,6045,160],{"className":6046},[159],[33,6048,6050],{"className":6049},[101],[33,6051,6053],{"className":6052,"style":167},[105],[33,6054],{},[33,6056],{"className":6057},[173,89],[33,6059,6061,6067,6118],{"className":6060},[84],[33,6062,6064],{"className":6063,"style":1071},[88,1070],[33,6065,1026],{"className":6066},[1075,1076],[33,6068,6070],{"className":6069},[52],[33,6071,6073,6110],{"className":6072},[96,97],[33,6074,6076,6107],{"className":6075},[101],[33,6077,6079,6093],{"className":6078,"style":4498},[105],[33,6080,6081,6084],{"style":4501},[33,6082],{"className":6083,"style":1171},[113],[33,6085,6087],{"className":6086},[118,119,120,121],[33,6088,6090],{"className":6089},[84,121],[33,6091,61],{"className":6092},[84,121],[33,6094,6095,6098],{"style":4516},[33,6096],{"className":6097,"style":1171},[113],[33,6099,6101],{"className":6100},[118,119,120,121],[33,6102,6104],{"className":6103},[84,121],[33,6105,1046],{"className":6106},[84,121],[33,6108,160],{"className":6109},[159],[33,6111,6113],{"className":6112},[101],[33,6114,6116],{"className":6115,"style":167},[105],[33,6117],{},[33,6119,6121],{"className":6120,"style":1071},[173,1070],[33,6122,1035],{"className":6123},[1075,1076],[33,6125],{"className":6126,"style":365},[364],[33,6128,276],{"className":6129},[369],[33,6131],{"className":6132,"style":365},[364],[33,6134,6136,6139],{"className":6135},[75],[33,6137],{"className":6138,"style":379},[79],[33,6140,3441],{"className":6141},[84],[15,6143,6144,6145,6148],{},"Exacto. Los números de Catalan cuentan precisamente cuántas secuencias de lanzamientos terminan ",[237,6146,6147],{},"por primera vez"," con más caras que cruces en cada longitud posible.",[559,6150,6152],{"id":6151},"la-conexión-con-el-triángulo-de-tartaglia","La conexión con el triángulo de Tartaglia",[15,6154,6155,6156,6159,6160,6215,6216,6345,6346,6350,6351,174],{},"Hay otra forma de ver por qué aparecen los números de Catalan aquí. Pensemos en cada secuencia de lanzamientos como un ",[237,6157,6158],{},"camino"," en una retícula: cara sube un paso, cruz baja un paso. El número total de caminos de longitud ",[33,6161,6163,6182],{"className":6162},[36],[33,6164,6166],{"className":6165},[40],[42,6167,6168],{"xmlns":44},[46,6169,6170,6180],{},[49,6171,6172,6174,6176,6178],{},[59,6173,402],{},[55,6175,2195],{},[274,6177,1314],{},[59,6179,270],{},[63,6181,2202],{"encoding":65},[33,6183,6185,6206],{"className":6184,"ariaHidden":71},[70],[33,6186,6188,6191,6194,6197,6200,6203],{"className":6187},[75],[33,6189],{"className":6190,"style":1329},[79],[33,6192,402],{"className":6193},[84],[33,6195,2195],{"className":6196},[84,154],[33,6198],{"className":6199,"style":2221},[364],[33,6201,1314],{"className":6202},[2225],[33,6204],{"className":6205,"style":2221},[364],[33,6207,6209,6212],{"className":6208},[75],[33,6210],{"className":6211,"style":379},[79],[33,6213,270],{"className":6214},[84]," que terminan un paso por encima de cero es el coeficiente binomial ",[33,6217,6219,6255],{"className":6218},[36],[33,6220,6222],{"className":6221},[40],[42,6223,6224],{"xmlns":44},[46,6225,6226,6252],{},[49,6227,6228,6230,6250],{},[274,6229,1026],{"fence":71},[52,6231,6232,6242],{"linethickness":4319},[49,6233,6234,6236,6238,6240],{},[59,6235,402],{},[55,6237,2195],{},[274,6239,1314],{},[59,6241,270],{},[49,6243,6244,6246,6248],{},[55,6245,2195],{},[274,6247,1314],{},[59,6249,270],{},[274,6251,1035],{"fence":71},[63,6253,6254],{"encoding":65},"\\binom{2n+1}{n+1}",[33,6256,6258],{"className":6257,"ariaHidden":71},[70],[33,6259,6261,6264],{"className":6260},[75],[33,6262],{"className":6263,"style":4402},[79],[33,6265,6267,6273,6339],{"className":6266},[84],[33,6268,6270],{"className":6269,"style":1071},[88,1070],[33,6271,1026],{"className":6272},[1075,1076],[33,6274,6276],{"className":6275},[52],[33,6277,6279,6331],{"className":6278},[96,97],[33,6280,6282,6328],{"className":6281},[101],[33,6283,6285,6305],{"className":6284,"style":4498},[105],[33,6286,6287,6290],{"style":4501},[33,6288],{"className":6289,"style":1171},[113],[33,6291,6293],{"className":6292},[118,119,120,121],[33,6294,6296,6299,6302],{"className":6295},[84,121],[33,6297,2195],{"className":6298},[84,154,121],[33,6300,1314],{"className":6301},[2225,121],[33,6303,270],{"className":6304},[84,121],[33,6306,6307,6310],{"style":4516},[33,6308],{"className":6309,"style":1171},[113],[33,6311,6313],{"className":6312},[118,119,120,121],[33,6314,6316,6319,6322,6325],{"className":6315},[84,121],[33,6317,402],{"className":6318},[84,121],[33,6320,2195],{"className":6321},[84,154,121],[33,6323,1314],{"className":6324},[2225,121],[33,6326,270],{"className":6327},[84,121],[33,6329,160],{"className":6330},[159],[33,6332,6334],{"className":6333},[101],[33,6335,6337],{"className":6336,"style":3734},[105],[33,6338],{},[33,6340,6342],{"className":6341,"style":1071},[173,1070],[33,6343,1035],{"className":6344},[1075,1076]," — una entrada del triángulo de Tartaglia. Los números de Catalan cuentan sólo los caminos que nunca cruzan por encima de cero antes del paso final. Son, literalmente, una selección filtrada dentro del triángulo — la misma herramienta que usamos en el ",[19,6347,6349],{"href":219,"rel":6348},[23],"artículo anterior"," para contar combinaciones de monedas, ahora con una restricción adicional. Si quieres explorar esta conexión en más detalle, Matt Parker la explica de forma brillante en ",[19,6352,6354],{"href":28,"rel":6353},[23],"su vídeo",[559,6356,6358],{"id":6357},"la-serie-infinita","La serie infinita",[15,6360,6361,6362,6417],{},"Volvamos a nuestra tabla. La contribución de las secuencias de longitud ",[33,6363,6365,6384],{"className":6364},[36],[33,6366,6368],{"className":6367},[40],[42,6369,6370],{"xmlns":44},[46,6371,6372,6382],{},[49,6373,6374,6376,6378,6380],{},[59,6375,402],{},[55,6377,2195],{},[274,6379,1314],{},[59,6381,270],{},[63,6383,2202],{"encoding":65},[33,6385,6387,6408],{"className":6386,"ariaHidden":71},[70],[33,6388,6390,6393,6396,6399,6402,6405],{"className":6389},[75],[33,6391],{"className":6392,"style":1329},[79],[33,6394,402],{"className":6395},[84],[33,6397,2195],{"className":6398},[84,154],[33,6400],{"className":6401,"style":2221},[364],[33,6403,1314],{"className":6404},[2225],[33,6406],{"className":6407,"style":2221},[364],[33,6409,6411,6414],{"className":6410},[75],[33,6412],{"className":6413,"style":379},[79],[33,6415,270],{"className":6416},[84]," al promedio total 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\\frac{C_n}{2^{2n+1}} \\cdot \\frac{n+1}{2n+1}",[33,7367,7369,7390,7632],{"className":7368,"ariaHidden":71},[70],[33,7370,7372,7375,7381,7384,7387],{"className":7371},[75],[33,7373],{"className":7374,"style":7024},[79],[33,7376,7378],{"className":7377},[84,623],[33,7379,7297],{"className":7380},[84],[33,7382],{"className":7383,"style":365},[364],[33,7385,276],{"className":7386},[369],[33,7388],{"className":7389,"style":365},[364],[33,7391,7393,7397,7468,7471,7623,7626,7629],{"className":7392},[75],[33,7394],{"className":7395,"style":7396},[79],"height:1.2334em;vertical-align:-0.345em;",[33,7398,7401,7407],{"className":7399},[7400],"mop",[33,7402,7305],{"className":7403,"style":7406},[7400,7404,7405],"op-symbol","small-op","position:relative;top:0em;",[33,7408,7410],{"className":7409},[1154],[33,7411,7413,7459],{"className":7412},[96,97],[33,7414,7416,7456],{"className":7415},[101],[33,7417,7420,7441],{"className":7418,"style":7419},[105],"height:0.8043em;",[33,7421,7423,7426],{"style":7422},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[33,7424],{"className":7425,"style":1171},[113],[33,7427,7429],{"className":7428},[118,119,120,121],[33,7430,7432,7435,7438],{"className":7431},[84,121],[33,7433,2195],{"className":7434},[84,154,121],[33,7436,276],{"className":7437},[369,121],[33,7439,515],{"className":7440},[84,121],[33,7442,7444,7447],{"style":7443},"top:-3.2029em;margin-right:0.05em;",[33,7445],{"className":7446,"style":1171},[113],[33,7448,7450],{"className":7449},[118,119,120,121],[33,7451,7453],{"className":7452},[84,121],[33,7454,7317],{"className":7455},[84,121],[33,7457,160],{"className":7458},[159],[33,7460,7462],{"className":7461},[101],[33,7463,7466],{"className":7464,"style":7465},[105],"height:0.2997em;",[33,7467],{},[33,7469],{"className":7470,"style":1450},[364],[33,7472,7474,7477,7620],{"className":7473},[84],[33,7475],{"className":7476},[88,89],[33,7478,7480],{"className":7479},[52],[33,7481,7483,7612],{"className":7482},[96,97],[33,7484,7486,7609],{"className":7485},[101],[33,7487,7490,7546,7554],{"className":7488,"style":7489},[105],"height:0.8884em;",[33,7491,7492,7495],{"style":109},[33,7493],{"className":7494,"style":114},[113],[33,7496,7498],{"className":7497},[118,119,120,121],[33,7499,7501],{"className":7500},[84,121],[33,7502,7504,7507],{"className":7503},[84,121],[33,7505,402],{"className":7506},[84,121],[33,7508,7510],{"className":7509},[1154],[33,7511,7513],{"className":7512},[96],[33,7514,7516],{"className":7515},[101],[33,7517,7520],{"className":7518,"style":7519},[105],"height:0.7463em;",[33,7521,7523,7527],{"style":7522},"top:-2.786em;margin-right:0.0714em;",[33,7524],{"className":7525,"style":7526},[113],"height:2.5em;",[33,7528,7531],{"className":7529},[118,7530,1076,121],"reset-size3",[33,7532,7534,7537,7540,7543],{"className":7533},[84,121],[33,7535,402],{"className":7536},[84,121],[33,7538,2195],{"className":7539},[84,154,121],[33,7541,1314],{"className":7542},[2225,121],[33,7544,270],{"className":7545},[84,121],[33,7547,7548,7551],{"style":130},[33,7549],{"className":7550,"style":114},[113],[33,7552],{"className":7553,"style":138},[137],[33,7555,7557,7560],{"style":7556},"top:-3.4101em;",[33,7558],{"className":7559,"style":114},[113],[33,7561,7563],{"className":7562},[118,119,120,121],[33,7564,7566],{"className":7565},[84,121],[33,7567,7569,7572],{"className":7568},[84,121],[33,7570,4296],{"className":7571,"style":4349},[84,154,121],[33,7573,7575],{"className":7574},[1154],[33,7576,7578,7600],{"className":7577},[96,97],[33,7579,7581,7597],{"className":7580},[101],[33,7582,7585],{"className":7583,"style":7584},[105],"height:0.1645em;",[33,7586,7588,7591],{"style":7587},"top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;",[33,7589],{"className":7590,"style":7526},[113],[33,7592,7594],{"className":7593},[118,7530,1076,121],[33,7595,2195],{"className":7596},[84,154,121],[33,7598,160],{"className":7599},[159],[33,7601,7603],{"className":7602},[101],[33,7604,7607],{"className":7605,"style":7606},[105],"height:0.143em;",[33,7608],{},[33,7610,160],{"className":7611},[159],[33,7613,7615],{"className":7614},[101],[33,7616,7618],{"className":7617,"style":167},[105],[33,7619],{},[33,7621],{"className":7622},[173,89],[33,7624],{"className":7625,"style":2221},[364],[33,7627,7342],{"className":7628},[2225],[33,7630],{"className":7631,"style":2221},[364],[33,7633,7635,7638],{"className":7634},[75],[33,7636],{"className":7637,"style":3655},[79],[33,7639,7641,7644,7718],{"className":7640},[84],[33,7642],{"className":7643},[88,89],[33,7645,7647],{"className":7646},[52],[33,7648,7650,7710],{"className":7649},[96,97],[33,7651,7653,7707],{"className":7652},[101],[33,7654,7656,7679,7687],{"className":7655,"style":310},[105],[33,7657,7658,7661],{"style":109},[33,7659],{"className":7660,"style":114},[113],[33,7662,7664],{"className":7663},[118,119,120,121],[33,7665,7667,7670,7673,7676],{"className":7666},[84,121],[33,7668,402],{"className":7669},[84,121],[33,7671,2195],{"className":7672},[84,154,121],[33,7674,1314],{"className":7675},[2225,121],[33,7677,270],{"className":7678},[84,121],[33,7680,7681,7684],{"style":130},[33,7682],{"className":7683,"style":114},[113],[33,7685],{"className":7686,"style":138},[137],[33,7688,7689,7692],{"style":141},[33,7690],{"className":7691,"style":114},[113],[33,7693,7695],{"className":7694},[118,119,120,121],[33,7696,7698,7701,7704],{"className":7697},[84,121],[33,7699,2195],{"className":7700},[84,154,121],[33,7702,1314],{"className":7703},[2225,121],[33,7705,270],{"className":7706},[84,121],[33,7708,160],{"className":7709},[159],[33,7711,7713],{"className":7712},[101],[33,7714,7716],{"className":7715,"style":3734},[105],[33,7717],{},[33,7719],{"className":7720},[173,89],[15,7722,7723],{},"Calculemos los primeros términos:",[2166,7725,7726,7767],{},[2169,7727,7728],{},[2172,7729,7730,7760,7764],{},[2175,7731,7732],{"align":2177},[33,7733,7735,7748],{"className":7734},[36],[33,7736,7738],{"className":7737},[40],[42,7739,7740],{"xmlns":44},[46,7741,7742,7746],{},[49,7743,7744],{},[55,7745,2195],{},[63,7747,2195],{"encoding":65},[33,7749,7751],{"className":7750,"ariaHidden":71},[70],[33,7752,7754,7757],{"className":7753},[75],[33,7755],{"className":7756,"style":204},[79],[33,7758,2195],{"className":7759},[84,154],[2175,7761,7763],{"align":7762},"left","Contribución",[2175,7765,7766],{"align":7762},"Acumulado",[2248,7768,7769,7990,8283,8575,8867],{},[2172,7770,7771,7773,7948],{},[2253,7772,515],{"align":2177},[2253,7774,7775],{"align":7762},[33,7776,7778,7808],{"className":7777},[36],[33,7779,7781],{"className":7780},[40],[42,7782,7783],{"xmlns":44},[46,7784,7785,7805],{},[49,7786,7787,7793,7795,7797,7799,7801,7803],{},[52,7788,7789,7791],{},[59,7790,270],{},[59,7792,402],{},[274,7794,6470],{},[59,7796,270],{},[274,7798,6470],{},[59,7800,270],{},[274,7802,276],{},[59,7804,498],{},[63,7806,7807],{"encoding":65},"\\frac{1}{2} \\times 1 \\times 1 = 0{,}5",[33,7809,7811,7894,7912,7930],{"className":7810,"ariaHidden":71},[70],[33,7812,7814,7817,7885,7888,7891],{"className":7813},[75],[33,7815],{"className":7816,"style":291},[79],[33,7818,7820,7823,7882],{"className":7819},[84],[33,7821],{"className":7822},[88,89],[33,7824,7826],{"className":7825},[52],[33,7827,7829,7874],{"className":7828},[96,97],[33,7830,7832,7871],{"className":7831},[101],[33,7833,7835,7849,7857],{"className":7834,"style":310},[105],[33,7836,7837,7840],{"style":109},[33,7838],{"className":7839,"style":114},[113],[33,7841,7843],{"className":7842},[118,119,120,121],[33,7844,7846],{"className":7845},[84,121],[33,7847,402],{"className":7848},[84,121],[33,7850,7851,7854],{"style":130},[33,7852],{"className":7853,"style":114},[113],[33,7855],{"className":7856,"style":138},[137],[33,7858,7859,7862],{"style":141},[33,7860],{"className":7861,"style":114},[113],[33,7863,7865],{"className":7864},[118,119,120,121],[33,7866,7868],{"className":7867},[84,121],[33,7869,270],{"className":7870},[84,121],[33,7872,160],{"className":7873},[159],[33,7875,7877],{"className":7876},[101],[33,7878,7880],{"className":7879,"style":167},[105],[33,7881],{},[33,7883],{"className":7884},[173,89],[33,7886],{"className":7887,"style":2221},[364],[33,7889,6470],{"className":7890},[2225],[33,7892],{"className":7893,"style":2221},[364],[33,7895,7897,7900,7903,7906,7909],{"className":7896},[75],[33,7898],{"className":7899,"style":1329},[79],[33,7901,270],{"className":7902},[84],[33,7904],{"className":7905,"style":2221},[364],[33,7907,6470],{"className":7908},[2225],[33,7910],{"className":7911,"style":2221},[364],[33,7913,7915,7918,7921,7924,7927],{"className":7914},[75],[33,7916],{"className":7917,"style":379},[79],[33,7919,270],{"className":7920},[84],[33,7922],{"className":7923,"style":365},[364],[33,7925,276],{"className":7926},[369],[33,7928],{"className":7929,"style":365},[364],[33,7931,7933,7936,7939,7945],{"className":7932},[75],[33,7934],{"className":7935,"style":511},[79],[33,7937,515],{"className":7938},[84],[33,7940,7942],{"className":7941},[84],[33,7943,523],{"className":7944},[522],[33,7946,527],{"className":7947},[84],[2253,7949,7950],{"align":7762},[33,7951,7953,7968],{"className":7952},[36],[33,7954,7956],{"className":7955},[40],[42,7957,7958],{"xmlns":44},[46,7959,7960,7965],{},[49,7961,7962],{},[59,7963,7964],{},"0,5000",[63,7966,7967],{"encoding":65},"0{,}5000",[33,7969,7971],{"className":7970,"ariaHidden":71},[70],[33,7972,7974,7977,7980,7986],{"className":7973},[75],[33,7975],{"className":7976,"style":511},[79],[33,7978,515],{"className":7979},[84],[33,7981,7983],{"className":7982},[84],[33,7984,523],{"className":7985},[522],[33,7987,7989],{"className":7988},[84],"5000",[2172,7991,7992,7994,8241],{},[2253,7993,270],{"align":2177},[2253,7995,7996],{"align":7762},[33,7997,7999,8035],{"className":7998},[36],[33,8000,8002],{"className":8001},[40],[42,8003,8004],{"xmlns":44},[46,8005,8006,8032],{},[49,8007,8008,8014,8016,8018,8020,8026,8029],{},[52,8009,8010,8012],{},[59,8011,270],{},[59,8013,1046],{},[274,8015,6470],{},[59,8017,270],{},[274,8019,6470],{},[52,8021,8022,8024],{},[59,8023,402],{},[59,8025,925],{},[274,8027,8028],{},"≈",[59,8030,8031],{},"0,0833",[63,8033,8034],{"encoding":65},"\\frac{1}{8} \\times 1 \\times \\frac{2}{3} \\approx 0{,}0833",[33,8036,8038,8121,8139,8222],{"className":8037,"ariaHidden":71},[70],[33,8039,8041,8044,8112,8115,8118],{"className":8040},[75],[33,8042],{"className":8043,"style":291},[79],[33,8045,8047,8050,8109],{"className":8046},[84],[33,8048],{"className":8049},[88,89],[33,8051,8053],{"className":8052},[52],[33,8054,8056,8101],{"className":8055},[96,97],[33,8057,8059,8098],{"className":8058},[101],[33,8060,8062,8076,8084],{"className":8061,"style":310},[105],[33,8063,8064,8067],{"style":109},[33,8065],{"className":8066,"style":114},[113],[33,8068,8070],{"className":8069},[118,119,120,121],[33,8071,8073],{"className":8072},[84,121],[33,8074,1046],{"className":8075},[84,121],[33,8077,8078,8081],{"style":130},[33,8079],{"className":8080,"style":114},[113],[33,8082],{"className":8083,"style":138},[137],[33,8085,8086,8089],{"style":141},[33,8087],{"className":8088,"style":114},[113],[33,8090,8092],{"className":8091},[118,119,120,121],[33,8093,8095],{"className":8094},[84,121],[33,8096,270],{"className":8097},[84,121],[33,8099,160],{"className":8100},[159],[33,8102,8104],{"className":8103},[101],[33,8105,8107],{"className":8106,"style":167},[105],[33,8108],{},[33,8110],{"className":8111},[173,89],[33,8113],{"className":8114,"style":2221},[364],[33,8116,6470],{"className":8117},[2225],[33,8119],{"className":8120,"style":2221},[364],[33,8122,8124,8127,8130,8133,8136],{"className":8123},[75],[33,8125],{"className":8126,"style":1329},[79],[33,8128,270],{"className":8129},[84],[33,8131],{"className":8132,"style":2221},[364],[33,8134,6470],{"className":8135},[2225],[33,8137],{"className":8138,"style":2221},[364],[33,8140,8142,8145,8213,8216,8219],{"className":8141},[75],[33,8143],{"className":8144,"style":291},[79],[33,8146,8148,8151,8210],{"className":8147},[84],[33,8149],{"className":8150},[88,89],[33,8152,8154],{"className":8153},[52],[33,8155,8157,8202],{"className":8156},[96,97],[33,8158,8160,8199],{"className":8159},[101],[33,8161,8163,8177,8185],{"className":8162,"style":310},[105],[33,8164,8165,8168],{"style":109},[33,8166],{"className":8167,"style":114},[113],[33,8169,8171],{"className":8170},[118,119,120,121],[33,8172,8174],{"className":8173},[84,121],[33,8175,925],{"className":8176},[84,121],[33,8178,8179,8182],{"style":130},[33,8180],{"className":8181,"style":114},[113],[33,8183],{"className":8184,"style":138},[137],[33,8186,8187,8190],{"style":141},[33,8188],{"className":8189,"style":114},[113],[33,8191,8193],{"className":8192},[118,119,120,121],[33,8194,8196],{"className":8195},[84,121],[33,8197,402],{"className":8198},[84,121],[33,8200,160],{"className":8201},[159],[33,8203,8205],{"className":8204},[101],[33,8206,8208],{"className":8207,"style":167},[105],[33,8209],{},[33,8211],{"className":8212},[173,89],[33,8214],{"className":8215,"style":365},[364],[33,8217,8028],{"className":8218},[369],[33,8220],{"className":8221,"style":365},[364],[33,8223,8225,8228,8231,8237],{"className":8224},[75],[33,8226],{"className":8227,"style":511},[79],[33,8229,515],{"className":8230},[84],[33,8232,8234],{"className":8233},[84],[33,8235,523],{"className":8236},[522],[33,8238,8240],{"className":8239},[84],"0833",[2253,8242,8243],{"align":7762},[33,8244,8246,8261],{"className":8245},[36],[33,8247,8249],{"className":8248},[40],[42,8250,8251],{"xmlns":44},[46,8252,8253,8258],{},[49,8254,8255],{},[59,8256,8257],{},"0,5833",[63,8259,8260],{"encoding":65},"0{,}5833",[33,8262,8264],{"className":8263,"ariaHidden":71},[70],[33,8265,8267,8270,8273,8279],{"className":8266},[75],[33,8268],{"className":8269,"style":511},[79],[33,8271,515],{"className":8272},[84],[33,8274,8276],{"className":8275},[84],[33,8277,523],{"className":8278},[522],[33,8280,8282],{"className":8281},[84],"5833",[2172,8284,8285,8287,8533],{},[2253,8286,402],{"align":2177},[2253,8288,8289],{"align":7762},[33,8290,8292,8327],{"className":8291},[36],[33,8293,8295],{"className":8294},[40],[42,8296,8297],{"xmlns":44},[46,8298,8299,8324],{},[49,8300,8301,8307,8309,8311,8313,8319,8321],{},[52,8302,8303,8305],{},[59,8304,270],{},[59,8306,1845],{},[274,8308,6470],{},[59,8310,402],{},[274,8312,6470],{},[52,8314,8315,8317],{},[59,8316,925],{},[59,8318,527],{},[274,8320,276],{},[59,8322,8323],{},"0,0375",[63,8325,8326],{"encoding":65},"\\frac{1}{32} \\times 2 \\times \\frac{3}{5} = 0{,}0375",[33,8328,8330,8413,8431,8514],{"className":8329,"ariaHidden":71},[70],[33,8331,8333,8336,8404,8407,8410],{"className":8332},[75],[33,8334],{"className":8335,"style":291},[79],[33,8337,8339,8342,8401],{"className":8338},[84],[33,8340],{"className":8341},[88,89],[33,8343,8345],{"className":8344},[52],[33,8346,8348,8393],{"className":8347},[96,97],[33,8349,8351,8390],{"className":8350},[101],[33,8352,8354,8368,8376],{"className":8353,"style":310},[105],[33,8355,8356,8359],{"style":109},[33,8357],{"className":8358,"style":114},[113],[33,8360,8362],{"className":8361},[118,119,120,121],[33,8363,8365],{"className":8364},[84,121],[33,8366,1845],{"className":8367},[84,121],[33,8369,8370,8373],{"style":130},[33,8371],{"className":8372,"style":114},[113],[33,8374],{"className":8375,"style":138},[137],[33,8377,8378,8381],{"style":141},[33,8379],{"className":8380,"style":114},[113],[33,8382,8384],{"className":8383},[118,119,120,121],[33,8385,8387],{"className":8386},[84,121],[33,8388,270],{"className":8389},[84,121],[33,8391,160],{"className":8392},[159],[33,8394,8396],{"className":8395},[101],[33,8397,8399],{"className":8398,"style":167},[105],[33,8400],{},[33,8402],{"className":8403},[173,89],[33,8405],{"className":8406,"style":2221},[364],[33,8408,6470],{"className":8409},[2225],[33,8411],{"className":8412,"style":2221},[364],[33,8414,8416,8419,8422,8425,8428],{"className":8415},[75],[33,8417],{"className":8418,"style":1329},[79],[33,8420,402],{"className":8421},[84],[33,8423],{"className":8424,"style":2221},[364],[33,8426,6470],{"className":8427},[2225],[33,8429],{"className":8430,"style":2221},[364],[33,8432,8434,8437,8505,8508,8511],{"className":8433},[75],[33,8435],{"className":8436,"style":291},[79],[33,8438,8440,8443,8502],{"className":8439},[84],[33,8441],{"className":8442},[88,89],[33,8444,8446],{"className":8445},[52],[33,8447,8449,8494],{"className":8448},[96,97],[33,8450,8452,8491],{"className":8451},[101],[33,8453,8455,8469,8477],{"className":8454,"style":310},[105],[33,8456,8457,8460],{"style":109},[33,8458],{"className":8459,"style":114},[113],[33,8461,8463],{"className":8462},[118,119,120,121],[33,8464,8466],{"className":8465},[84,121],[33,8467,527],{"className":8468},[84,121],[33,8470,8471,8474],{"style":130},[33,8472],{"className":8473,"style":114},[113],[33,8475],{"className":8476,"style":138},[137],[33,8478,8479,8482],{"style":141},[33,8480],{"className":8481,"style":114},[113],[33,8483,8485],{"className":8484},[118,119,120,121],[33,8486,8488],{"className":8487},[84,121],[33,8489,925],{"className":8490},[84,121],[33,8492,160],{"className":8493},[159],[33,8495,8497],{"className":8496},[101],[33,8498,8500],{"className":8499,"style":167},[105],[33,8501],{},[33,8503],{"className":8504},[173,89],[33,8506],{"className":8507,"style":365},[364],[33,8509,276],{"className":8510},[369],[33,8512],{"className":8513,"style":365},[364],[33,8515,8517,8520,8523,8529],{"className":8516},[75],[33,8518],{"className":8519,"style":511},[79],[33,8521,515],{"className":8522},[84],[33,8524,8526],{"className":8525},[84],[33,8527,523],{"className":8528},[522],[33,8530,8532],{"className":8531},[84],"0375",[2253,8534,8535],{"align":7762},[33,8536,8538,8553],{"className":8537},[36],[33,8539,8541],{"className":8540},[40],[42,8542,8543],{"xmlns":44},[46,8544,8545,8550],{},[49,8546,8547],{},[59,8548,8549],{},"0,6208",[63,8551,8552],{"encoding":65},"0{,}6208",[33,8554,8556],{"className":8555,"ariaHidden":71},[70],[33,8557,8559,8562,8565,8571],{"className":8558},[75],[33,8560],{"className":8561,"style":511},[79],[33,8563,515],{"className":8564},[84],[33,8566,8568],{"className":8567},[84],[33,8569,523],{"className":8570},[522],[33,8572,8574],{"className":8573},[84],"6208",[2172,8576,8577,8579,8825],{},[2253,8578,925],{"align":2177},[2253,8580,8581],{"align":7762},[33,8582,8584,8619],{"className":8583},[36],[33,8585,8587],{"className":8586},[40],[42,8588,8589],{"xmlns":44},[46,8590,8591,8616],{},[49,8592,8593,8599,8601,8603,8605,8611,8613],{},[52,8594,8595,8597],{},[59,8596,270],{},[59,8598,3201],{},[274,8600,6470],{},[59,8602,527],{},[274,8604,6470],{},[52,8606,8607,8609],{},[59,8608,61],{},[59,8610,3098],{},[274,8612,8028],{},[59,8614,8615],{},"0,0223",[63,8617,8618],{"encoding":65},"\\frac{1}{128} \\times 5 \\times \\frac{4}{7} \\approx 0{,}0223",[33,8620,8622,8705,8723,8806],{"className":8621,"ariaHidden":71},[70],[33,8623,8625,8628,8696,8699,8702],{"className":8624},[75],[33,8626],{"className":8627,"style":291},[79],[33,8629,8631,8634,8693],{"className":8630},[84],[33,8632],{"className":8633},[88,89],[33,8635,8637],{"className":8636},[52],[33,8638,8640,8685],{"className":8639},[96,97],[33,8641,8643,8682],{"className":8642},[101],[33,8644,8646,8660,8668],{"className":8645,"style":310},[105],[33,8647,8648,8651],{"style":109},[33,8649],{"className":8650,"style":114},[113],[33,8652,8654],{"className":8653},[118,119,120,121],[33,8655,8657],{"className":8656},[84,121],[33,8658,3201],{"className":8659},[84,121],[33,8661,8662,8665],{"style":130},[33,8663],{"className":8664,"style":114},[113],[33,8666],{"className":8667,"style":138},[137],[33,8669,8670,8673],{"style":141},[33,8671],{"className":8672,"style":114},[113],[33,8674,8676],{"className":8675},[118,119,120,121],[33,8677,8679],{"className":8678},[84,121],[33,8680,270],{"className":8681},[84,121],[33,8683,160],{"className":8684},[159],[33,8686,8688],{"className":8687},[101],[33,8689,8691],{"className":8690,"style":167},[105],[33,8692],{},[33,8694],{"className":8695},[173,89],[33,8697],{"className":8698,"style":2221},[364],[33,8700,6470],{"className":8701},[2225],[33,8703],{"className":8704,"style":2221},[364],[33,8706,8708,8711,8714,8717,8720],{"className":8707},[75],[33,8709],{"className":8710,"style":1329},[79],[33,8712,527],{"className":8713},[84],[33,8715],{"className":8716,"style":2221},[364],[33,8718,6470],{"className":8719},[2225],[33,8721],{"className":8722,"style":2221},[364],[33,8724,8726,8729,8797,8800,8803],{"className":8725},[75],[33,8727],{"className":8728,"style":291},[79],[33,8730,8732,8735,8794],{"className":8731},[84],[33,8733],{"className":8734},[88,89],[33,8736,8738],{"className":8737},[52],[33,8739,8741,8786],{"className":8740},[96,97],[33,8742,8744,8783],{"className":8743},[101],[33,8745,8747,8761,8769],{"className":8746,"style":310},[105],[33,8748,8749,8752],{"style":109},[33,8750],{"className":8751,"style":114},[113],[33,8753,8755],{"className":8754},[118,119,120,121],[33,8756,8758],{"className":8757},[84,121],[33,8759,3098],{"className":8760},[84,121],[33,8762,8763,8766],{"style":130},[33,8764],{"className":8765,"style":114},[113],[33,8767],{"className":8768,"style":138},[137],[33,8770,8771,8774],{"style":141},[33,8772],{"className":8773,"style":114},[113],[33,8775,8777],{"className":8776},[118,119,120,121],[33,8778,8780],{"className":8779},[84,121],[33,8781,61],{"className":8782},[84,121],[33,8784,160],{"className":8785},[159],[33,8787,8789],{"className":8788},[101],[33,8790,8792],{"className":8791,"style":167},[105],[33,8793],{},[33,8795],{"className":8796},[173,89],[33,8798],{"className":8799,"style":365},[364],[33,8801,8028],{"className":8802},[369],[33,8804],{"className":8805,"style":365},[364],[33,8807,8809,8812,8815,8821],{"className":8808},[75],[33,8810],{"className":8811,"style":511},[79],[33,8813,515],{"className":8814},[84],[33,8816,8818],{"className":8817},[84],[33,8819,523],{"className":8820},[522],[33,8822,8824],{"className":8823},[84],"0223",[2253,8826,8827],{"align":7762},[33,8828,8830,8845],{"className":8829},[36],[33,8831,8833],{"className":8832},[40],[42,8834,8835],{"xmlns":44},[46,8836,8837,8842],{},[49,8838,8839],{},[59,8840,8841],{},"0,6431",[63,8843,8844],{"encoding":65},"0{,}6431",[33,8846,8848],{"className":8847,"ariaHidden":71},[70],[33,8849,8851,8854,8857,8863],{"className":8850},[75],[33,8852],{"className":8853,"style":511},[79],[33,8855,515],{"className":8856},[84],[33,8858,8860],{"className":8859},[84],[33,8861,523],{"className":8862},[522],[33,8864,8866],{"className":8865},[84],"6431",[2172,8868,8869,8871,9117],{},[2253,8870,61],{"align":2177},[2253,8872,8873],{"align":7762},[33,8874,8876,8911],{"className":8875},[36],[33,8877,8879],{"className":8878},[40],[42,8880,8881],{"xmlns":44},[46,8882,8883,8908],{},[49,8884,8885,8891,8893,8895,8897,8903,8905],{},[52,8886,8887,8889],{},[59,8888,270],{},[59,8890,3462],{},[274,8892,6470],{},[59,8894,3441],{},[274,8896,6470],{},[52,8898,8899,8901],{},[59,8900,527],{},[59,8902,3358],{},[274,8904,8028],{},[59,8906,8907],{},"0,0152",[63,8909,8910],{"encoding":65},"\\frac{1}{512} \\times 14 \\times \\frac{5}{9} \\approx 0{,}0152",[33,8912,8914,8997,9015,9098],{"className":8913,"ariaHidden":71},[70],[33,8915,8917,8920,8988,8991,8994],{"className":8916},[75],[33,8918],{"className":8919,"style":291},[79],[33,8921,8923,8926,8985],{"className":8922},[84],[33,8924],{"className":8925},[88,89],[33,8927,8929],{"className":8928},[52],[33,8930,8932,8977],{"className":8931},[96,97],[33,8933,8935,8974],{"className":8934},[101],[33,8936,8938,8952,8960],{"className":8937,"style":310},[105],[33,8939,8940,8943],{"style":109},[33,8941],{"className":8942,"style":114},[113],[33,8944,8946],{"className":8945},[118,119,120,121],[33,8947,8949],{"className":8948},[84,121],[33,8950,3462],{"className":8951},[84,121],[33,8953,8954,8957],{"style":130},[33,8955],{"className":8956,"style":114},[113],[33,8958],{"className":8959,"style":138},[137],[33,8961,8962,8965],{"style":141},[33,8963],{"className":8964,"style":114},[113],[33,8966,8968],{"className":8967},[118,119,120,121],[33,8969,8971],{"className":8970},[84,121],[33,8972,270],{"className":8973},[84,121],[33,8975,160],{"className":8976},[159],[33,8978,8980],{"className":8979},[101],[33,8981,8983],{"className":8982,"style":167},[105],[33,8984],{},[33,8986],{"className":8987},[173,89],[33,8989],{"className":8990,"style":2221},[364],[33,8992,6470],{"className":8993},[2225],[33,8995],{"className":8996,"style":2221},[364],[33,8998,9000,9003,9006,9009,9012],{"className":8999},[75],[33,9001],{"className":9002,"style":1329},[79],[33,9004,3441],{"className":9005},[84],[33,9007],{"className":9008,"style":2221},[364],[33,9010,6470],{"className":9011},[2225],[33,9013],{"className":9014,"style":2221},[364],[33,9016,9018,9021,9089,9092,9095],{"className":9017},[75],[33,9019],{"className":9020,"style":291},[79],[33,9022,9024,9027,9086],{"className":9023},[84],[33,9025],{"className":9026},[88,89],[33,9028,9030],{"className":9029},[52],[33,9031,9033,9078],{"className":9032},[96,97],[33,9034,9036,9075],{"className":9035},[101],[33,9037,9039,9053,9061],{"className":9038,"style":310},[105],[33,9040,9041,9044],{"style":109},[33,9042],{"className":9043,"style":114},[113],[33,9045,9047],{"className":9046},[118,119,120,121],[33,9048,9050],{"className":9049},[84,121],[33,9051,3358],{"className":9052},[84,121],[33,9054,9055,9058],{"style":130},[33,9056],{"className":9057,"style":114},[113],[33,9059],{"className":9060,"style":138},[137],[33,9062,9063,9066],{"style":141},[33,9064],{"className":9065,"style":114},[113],[33,9067,9069],{"className":9068},[118,119,120,121],[33,9070,9072],{"className":9071},[84,121],[33,9073,527],{"className":9074},[84,121],[33,9076,160],{"className":9077},[159],[33,9079,9081],{"className":9080},[101],[33,9082,9084],{"className":9083,"style":167},[105],[33,9085],{},[33,9087],{"className":9088},[173,89],[33,9090],{"className":9091,"style":365},[364],[33,9093,8028],{"className":9094},[369],[33,9096],{"className":9097,"style":365},[364],[33,9099,9101,9104,9107,9113],{"className":9100},[75],[33,9102],{"className":9103,"style":511},[79],[33,9105,515],{"className":9106},[84],[33,9108,9110],{"className":9109},[84],[33,9111,523],{"className":9112},[522],[33,9114,9116],{"className":9115},[84],"0152",[2253,9118,9119],{"align":7762},[33,9120,9122,9137],{"className":9121},[36],[33,9123,9125],{"className":9124},[40],[42,9126,9127],{"xmlns":44},[46,9128,9129,9134],{},[49,9130,9131],{},[59,9132,9133],{},"0,6583",[63,9135,9136],{"encoding":65},"0{,}6583",[33,9138,9140],{"className":9139,"ariaHidden":71},[70],[33,9141,9143,9146,9149,9155],{"className":9142},[75],[33,9144],{"className":9145,"style":511},[79],[33,9147,515],{"className":9148},[84],[33,9150,9152],{"className":9151},[84],[33,9153,523],{"className":9154},[522],[33,9156,9158],{"className":9157},[84],"6583",[15,9160,9161,9162,9199,9200,9331],{},"Con sólo cinco términos ya estamos en ",[33,9163,9165,9178],{"className":9164},[36],[33,9166,9168],{"className":9167},[40],[42,9169,9170],{"xmlns":44},[46,9171,9172,9176],{},[49,9173,9174],{},[59,9175,9133],{},[63,9177,9136],{"encoding":65},[33,9179,9181],{"className":9180,"ariaHidden":71},[70],[33,9182,9184,9187,9190,9196],{"className":9183},[75],[33,9185],{"className":9186,"style":511},[79],[33,9188,515],{"className":9189},[84],[33,9191,9193],{"className":9192},[84],[33,9194,523],{"className":9195},[522],[33,9197,9158],{"className":9198},[84],", acercándonos lentamente a ",[33,9201,9203,9226],{"className":9202},[36],[33,9204,9206],{"className":9205},[40],[42,9207,9208],{"xmlns":44},[46,9209,9210,9223],{},[49,9211,9212,9218,9220],{},[52,9213,9214,9216],{},[55,9215,57],{},[59,9217,61],{},[274,9219,8028],{},[59,9221,9222],{},"0,7854",[63,9224,9225],{"encoding":65},"\\frac{\\pi}{4} \\approx 0{,}7854",[33,9227,9229,9312],{"className":9228,"ariaHidden":71},[70],[33,9230,9232,9235,9303,9306,9309],{"className":9231},[75],[33,9233],{"className":9234,"style":80},[79],[33,9236,9238,9241,9300],{"className":9237},[84],[33,9239],{"className":9240},[88,89],[33,9242,9244],{"className":9243},[52],[33,9245,9247,9292],{"className":9246},[96,97],[33,9248,9250,9289],{"className":9249},[101],[33,9251,9253,9267,9275],{"className":9252,"style":106},[105],[33,9254,9255,9258],{"style":109},[33,9256],{"className":9257,"style":114},[113],[33,9259,9261],{"className":9260},[118,119,120,121],[33,9262,9264],{"className":9263},[84,121],[33,9265,61],{"className":9266},[84,121],[33,9268,9269,9272],{"style":130},[33,9270],{"className":9271,"style":114},[113],[33,9273],{"className":9274,"style":138},[137],[33,9276,9277,9280],{"style":141},[33,9278],{"className":9279,"style":114},[113],[33,9281,9283],{"className":9282},[118,119,120,121],[33,9284,9286],{"className":9285},[84,121],[33,9287,57],{"className":9288,"style":155},[84,154,121],[33,9290,160],{"className":9291},[159],[33,9293,9295],{"className":9294},[101],[33,9296,9298],{"className":9297,"style":167},[105],[33,9299],{},[33,9301],{"className":9302},[173,89],[33,9304],{"className":9305,"style":365},[364],[33,9307,8028],{"className":9308},[369],[33,9310],{"className":9311,"style":365},[364],[33,9313,9315,9318,9321,9327],{"className":9314},[75],[33,9316],{"className":9317,"style":511},[79],[33,9319,515],{"className":9320},[84],[33,9322,9324],{"className":9323},[84],[33,9325,523],{"className":9326},[522],[33,9328,9330],{"className":9329},[84],"7854",". La convergencia es lenta, pero la dirección es inequívoca cuando el número de lanzamientos es suficientemente grande.",[559,9333,9335],{"id":9334},"la-aparición-del-arcoseno","La aparición del arcoseno",[15,9337,9338],{},"Aquí es donde ocurre la magia. Sustituyamos la fórmula de los números de Catalan en nuestra serie:",[15,9340,9342],{"className":9341},[590],[33,9343,9345,9392],{"className":9344},[36],[33,9346,9348],{"className":9347},[40],[42,9349,9350],{"xmlns":44},[46,9351,9352,9390],{},[49,9353,9354,9360,9362,9374],{},[4292,9355,9356,9358],{},[55,9357,4296],{},[55,9359,2195],{},[274,9361,276],{},[52,9363,9364,9366],{},[59,9365,270],{},[49,9367,9368,9370,9372],{},[55,9369,2195],{},[274,9371,1314],{},[59,9373,270],{},[49,9375,9376,9378,9388],{},[274,9377,1026],{"fence":71},[52,9379,9380,9386],{"linethickness":4319},[49,9381,9382,9384],{},[59,9383,402],{},[55,9385,2195],{},[55,9387,2195],{},[274,9389,1035],{"fence":71},[63,9391,4332],{"encoding":65},[33,9393,9395,9450],{"className":9394,"ariaHidden":71},[70],[33,9396,9398,9401,9441,9444,9447],{"className":9397},[75],[33,9399],{"className":9400,"style":4342},[79],[33,9402,9404,9407],{"className":9403},[84],[33,9405,4296],{"className":9406,"style":4349},[84,154],[33,9408,9410],{"className":9409},[1154],[33,9411,9413,9433],{"className":9412},[96,97],[33,9414,9416,9430],{"className":9415},[101],[33,9417,9419],{"className":9418,"style":4362},[105],[33,9420,9421,9424],{"style":4365},[33,9422],{"className":9423,"style":1171},[113],[33,9425,9427],{"className":9426},[118,119,120,121],[33,9428,2195],{"className":9429},[84,154,121],[33,9431,160],{"className":9432},[159],[33,9434,9436],{"className":9435},[101],[33,9437,9439],{"className":9438,"style":4384},[105],[33,9440],{},[33,9442],{"className":9443,"style":365},[364],[33,9445,276],{"className":9446},[369],[33,9448],{"className":9449,"style":365},[364],[33,9451,9453,9456,9530],{"className":9452},[75],[33,9454],{"className":9455,"style":4402},[79],[33,9457,9459,9462,9527],{"className":9458},[84],[33,9460],{"className":9461},[88,89],[33,9463,9465],{"className":9464},[52],[33,9466,9468,9519],{"className":9467},[96,97],[33,9469,9471,9516],{"className":9470},[101],[33,9472,9474,9494,9502],{"className":9473,"style":310},[105],[33,9475,9476,9479],{"style":109},[33,9477],{"className":9478,"style":114},[113],[33,9480,9482],{"className":9481},[118,119,120,121],[33,9483,9485,9488,9491],{"className":9484},[84,121],[33,9486,2195],{"className":9487},[84,154,121],[33,9489,1314],{"className":9490},[2225,121],[33,9492,270],{"className":9493},[84,121],[33,9495,9496,9499],{"style":130},[33,9497],{"className":9498,"style":114},[113],[33,9500],{"className":9501,"style":138},[137],[33,9503,9504,9507],{"style":141},[33,9505],{"className":9506,"style":114},[113],[33,9508,9510],{"className":9509},[118,119,120,121],[33,9511,9513],{"className":9512},[84,121],[33,9514,270],{"className":9515},[84,121],[33,9517,160],{"className":9518},[159],[33,9520,9522],{"className":9521},[101],[33,9523,9525],{"className":9524,"style":3734},[105],[33,9526],{},[33,9528],{"className":9529},[173,89],[33,9531,9533,9539,9593],{"className":9532},[84],[33,9534,9536],{"className":9535,"style":1071},[88,1070],[33,9537,1026],{"className":9538},[1075,1076],[33,9540,9542],{"className":9541},[52],[33,9543,9545,9585],{"className":9544},[96,97],[33,9546,9548,9582],{"className":9547},[101],[33,9549,9551,9565],{"className":9550,"style":4498},[105],[33,9552,9553,9556],{"style":4501},[33,9554],{"className":9555,"style":1171},[113],[33,9557,9559],{"className":9558},[118,119,120,121],[33,9560,9562],{"className":9561},[84,121],[33,9563,2195],{"className":9564},[84,154,121],[33,9566,9567,9570],{"style":4516},[33,9568],{"className":9569,"style":1171},[113],[33,9571,9573],{"className":9572},[118,119,120,121],[33,9574,9576,9579],{"className":9575},[84,121],[33,9577,402],{"className":9578},[84,121],[33,9580,2195],{"className":9581},[84,154,121],[33,9583,160],{"className":9584},[159],[33,9586,9588],{"className":9587},[101],[33,9589,9591],{"className":9590,"style":167},[105],[33,9592],{},[33,9594,9596],{"className":9595,"style":1071},[173,1070],[33,9597,1035],{"className":9598},[1075,1076],[15,9600,9601,9602,9672,9673,9799,9800,9863],{},"Al multiplicar 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por ",[33,9674,9676,9707],{"className":9675},[36],[33,9677,9679],{"className":9678},[40],[42,9680,9681],{"xmlns":44},[46,9682,9683,9705],{},[49,9684,9685],{},[52,9686,9687,9695],{},[49,9688,9689,9691,9693],{},[55,9690,2195],{},[274,9692,1314],{},[59,9694,270],{},[49,9696,9697,9699,9701,9703],{},[59,9698,402],{},[55,9700,2195],{},[274,9702,1314],{},[59,9704,270],{},[63,9706,3645],{"encoding":65},[33,9708,9710],{"className":9709,"ariaHidden":71},[70],[33,9711,9713,9716],{"className":9712},[75],[33,9714],{"className":9715,"style":3655},[79],[33,9717,9719,9722,9796],{"className":9718},[84],[33,9720],{"className":9721},[88,89],[33,9723,9725],{"className":9724},[52],[33,9726,9728,9788],{"className":9727},[96,97],[33,9729,9731,9785],{"className":9730},[101],[33,9732,9734,9757,9765],{"className":9733,"style":310},[105],[33,9735,9736,9739],{"style":109},[33,9737],{"className":9738,"style":114},[113],[33,9740,9742],{"className":9741},[118,119,120,121],[33,9743,9745,9748,9751,9754],{"className":9744},[84,121],[33,9746,402],{"className":9747},[84,121],[33,9749,2195],{"className":9750},[84,154,121],[33,9752,1314],{"className":9753},[2225,121],[33,9755,270],{"className":9756},[84,121],[33,9758,9759,9762],{"style":130},[33,9760],{"className":9761,"style":114},[113],[33,9763],{"className":9764,"style":138},[137],[33,9766,9767,9770],{"style":141},[33,9768],{"className":9769,"style":114},[113],[33,9771,9773],{"className":9772},[118,119,120,121],[33,9774,9776,9779,9782],{"className":9775},[84,121],[33,9777,2195],{"className":9778},[84,154,121],[33,9780,1314],{"className":9781},[2225,121],[33,9783,270],{"className":9784},[84,121],[33,9786,160],{"className":9787},[159],[33,9789,9791],{"className":9790},[101],[33,9792,9794],{"className":9793,"style":3734},[105],[33,9795],{},[33,9797],{"className":9798},[173,89],", el factor ",[33,9801,9803,9826],{"className":9802},[36],[33,9804,9806],{"className":9805},[40],[42,9807,9808],{"xmlns":44},[46,9809,9810,9823],{},[49,9811,9812,9815,9817,9819,9821],{},[274,9813,1026],{"stretchy":9814},"false",[55,9816,2195],{},[274,9818,1314],{},[59,9820,270],{},[274,9822,1035],{"stretchy":9814},[63,9824,9825],{"encoding":65},"(n+1)",[33,9827,9829,9851],{"className":9828,"ariaHidden":71},[70],[33,9830,9832,9836,9839,9842,9845,9848],{"className":9831},[75],[33,9833],{"className":9834,"style":9835},[79],"height:1em;vertical-align:-0.25em;",[33,9837,1026],{"className":9838},[88],[33,9840,2195],{"className":9841},[84,154],[33,9843],{"className":9844,"style":2221},[364],[33,9846,1314],{"className":9847},[2225],[33,9849],{"className":9850,"style":2221},[364],[33,9852,9854,9857,9860],{"className":9853},[75],[33,9855],{"className":9856,"style":9835},[79],[33,9858,270],{"className":9859},[84],[33,9861,1035],{"className":9862},[173]," se cancela limpiamente:",[15,9865,9867],{"className":9866},[590],[33,9868,9870,9962],{"className":9869},[36],[33,9871,9873],{"className":9872},[40],[42,9874,9875],{"xmlns":44},[46,9876,9877,9959],{},[49,9878,9879,9891,9907,9909,9929,9931],{},[52,9880,9881,9883],{},[59,9882,270],{},[49,9884,9885,9887,9889],{},[55,9886,2195],{},[274,9888,1314],{},[59,9890,270],{},[49,9892,9893,9895,9905],{},[274,9894,1026],{"fence":71},[52,9896,9897,9903],{"linethickness":4319},[49,9898,9899,9901],{},[59,9900,402],{},[55,9902,2195],{},[55,9904,2195],{},[274,9906,1035],{"fence":71},[274,9908,7342],{},[52,9910,9911,9919],{},[49,9912,9913,9915,9917],{},[55,9914,2195],{},[274,9916,1314],{},[59,9918,270],{},[49,9920,9921,9923,9925,9927],{},[59,9922,402],{},[55,9924,2195],{},[274,9926,1314],{},[59,9928,270],{},[274,9930,276],{},[52,9932,9933,9949],{},[49,9934,9935,9937,9947],{},[274,9936,1026],{"fence":71},[52,9938,9939,9945],{"linethickness":4319},[49,9940,9941,9943],{},[59,9942,402],{},[55,9944,2195],{},[55,9946,2195],{},[274,9948,1035],{"fence":71},[49,9950,9951,9953,9955,9957],{},[59,9952,402],{},[55,9954,2195],{},[274,9956,1314],{},[59,9958,270],{},[63,9960,9961],{"encoding":65},"\\frac{1}{n+1}\\binom{2n}{n} \\cdot \\frac{n+1}{2n+1} = \\frac{\\binom{2n}{n}}{2n+1}",[33,9963,9965,10123,10221],{"className":9964,"ariaHidden":71},[70],[33,9966,9968,9971,10045,10114,10117,10120],{"className":9967},[75],[33,9969],{"className":9970,"style":4402},[79],[33,9972,9974,9977,10042],{"className":9973},[84],[33,9975],{"className":9976},[88,89],[33,9978,9980],{"className":9979},[52],[33,9981,9983,10034],{"className":9982},[96,97],[33,9984,9986,10031],{"className":9985},[101],[33,9987,9989,10009,10017],{"className":9988,"style":310},[105],[33,9990,9991,9994],{"style":109},[33,9992],{"className":9993,"style":114},[113],[33,9995,9997],{"className":9996},[118,119,120,121],[33,9998,10000,10003,10006],{"className":9999},[84,121],[33,10001,2195],{"className":10002},[84,154,121],[33,10004,1314],{"className":10005},[2225,121],[33,10007,270],{"className":10008},[84,121],[33,10010,10011,10014],{"style":130},[33,10012],{"className":10013,"style":114},[113],[33,10015],{"className":10016,"style":138},[137],[33,10018,10019,10022],{"style":141},[33,10020],{"className":10021,"style":114},[113],[33,10023,10025],{"className":10024},[118,119,120,121],[33,10026,10028],{"className":10027},[84,121],[33,10029,270],{"className":10030},[84,121],[33,10032,160],{"className":10033},[159],[33,10035,10037],{"className":10036},[101],[33,10038,10040],{"className":10039,"style":3734},[105],[33,10041],{},[33,10043],{"className":10044},[173,89],[33,10046,10048,10054,10108],{"className":10047},[84],[33,10049,10051],{"className":10050,"style":1071},[88,1070],[33,10052,1026],{"className":10053},[1075,1076],[33,10055,10057],{"className":10056},[52],[33,10058,10060,10100],{"className":10059},[96,97],[33,10061,10063,10097],{"className":10062},[101],[33,10064,10066,10080],{"className":10065,"style":4498},[105],[33,10067,10068,10071],{"style":4501},[33,10069],{"className":10070,"style":1171},[113],[33,10072,10074],{"className":10073},[118,119,120,121],[33,10075,10077],{"className":10076},[84,121],[33,10078,2195],{"className":10079},[84,154,121],[33,10081,10082,10085],{"style":4516},[33,10083],{"className":10084,"style":1171},[113],[33,10086,10088],{"className":10087},[118,119,120,121],[33,10089,10091,10094],{"className":10090},[84,121],[33,10092,402],{"className":10093},[84,121],[33,10095,2195],{"className":10096},[84,154,121],[33,10098,160],{"className":10099},[159],[33,10101,10103],{"className":10102},[101],[33,10104,10106],{"className":10105,"style":167},[105],[33,10107],{},[33,10109,10111],{"className":10110,"style":1071},[173,1070],[33,10112,1035],{"className":10113},[1075,1076],[33,10115],{"className":10116,"style":2221},[364],[33,10118,7342],{"className":10119},[2225],[33,10121],{"className":10122,"style":2221},[364],[33,10124,10126,10129,10212,10215,10218],{"className":10125},[75],[33,10127],{"className":10128,"style":3655},[79],[33,10130,10132,10135,10209],{"className":10131},[84],[33,10133],{"className":10134},[88,89],[33,10136,10138],{"className":10137},[52],[33,10139,10141,10201],{"className":10140},[96,97],[33,10142,10144,10198],{"className":10143},[101],[33,10145,10147,10170,10178],{"className":10146,"style":310},[105],[33,10148,10149,10152],{"style":109},[33,10150],{"className":10151,"style":114},[113],[33,10153,10155],{"className":10154},[118,119,120,121],[33,10156,10158,10161,10164,10167],{"className":10157},[84,121],[33,10159,402],{"className":10160},[84,121],[33,10162,2195],{"className":10163},[84,154,121],[33,10165,1314],{"className":10166},[2225,121],[33,10168,270],{"className":10169},[84,121],[33,10171,10172,10175],{"style":130},[33,10173],{"className":10174,"style":114},[113],[33,10176],{"className":10177,"style":138},[137],[33,10179,10180,10183],{"style":141},[33,10181],{"className":10182,"style":114},[113],[33,10184,10186],{"className":10185},[118,119,120,121],[33,10187,10189,10192,10195],{"className":10188},[84,121],[33,10190,2195],{"className":10191},[84,154,121],[33,10193,1314],{"className":10194},[2225,121],[33,10196,270],{"className":10197},[84,121],[33,10199,160],{"className":10200},[159],[33,10202,10204],{"className":10203},[101],[33,10205,10207],{"className":10206,"style":3734},[105],[33,10208],{},[33,10210],{"className":10211},[173,89],[33,10213],{"className":10214,"style":365},[364],[33,10216,276],{"className":10217},[369],[33,10219],{"className":10220,"style":365},[364],[33,10222,10224,10228],{"className":10223},[75],[33,10225],{"className":10226,"style":10227},[79],"height:1.9133em;vertical-align:-0.4033em;",[33,10229,10231,10234,10382],{"className":10230},[84],[33,10232],{"className":10233},[88,89],[33,10235,10237],{"className":10236},[52],[33,10238,10240,10374],{"className":10239},[96,97],[33,10241,10243,10371],{"className":10242},[101],[33,10244,10247,10270,10278],{"className":10245,"style":10246},[105],"height:1.51em;",[33,10248,10249,10252],{"style":109},[33,10250],{"className":10251,"style":114},[113],[33,10253,10255],{"className":10254},[118,119,120,121],[33,10256,10258,10261,10264,10267],{"className":10257},[84,121],[33,10259,402],{"className":10260},[84,121],[33,10262,2195],{"className":10263},[84,154,121],[33,10265,1314],{"className":10266},[2225,121],[33,10268,270],{"className":10269},[84,121],[33,10271,10272,10275],{"style":130},[33,10273],{"className":10274,"style":114},[113],[33,10276],{"className":10277,"style":138},[137],[33,10279,10281,10284],{"style":10280},"top:-3.7125em;",[33,10282],{"className":10283,"style":114},[113],[33,10285,10287],{"className":10286},[118,119,120,121],[33,10288,10290],{"className":10289},[84,121],[33,10291,10293,10304,10362],{"className":10292},[84,121],[33,10294,10298],{"className":10295,"style":10297},[88,118,7530,10296,121,1070],"size6","top:0.075em;",[33,10299,10301],{"className":10300},[1075,1076,121],[33,10302,1026],{"className":10303},[121],[33,10305,10307],{"className":10306},[52],[33,10308,10310,10353],{"className":10309},[96,97],[33,10311,10313,10350],{"className":10312},[101],[33,10314,10317,10332],{"className":10315,"style":10316},[105],"height:0.9313em;",[33,10318,10320,10323],{"style":10319},"top:-2.156em;",[33,10321],{"className":10322,"style":7526},[113],[33,10324,10326],{"className":10325},[118,7530,1076,121],[33,10327,10329],{"className":10328},[84,121],[33,10330,2195],{"className":10331},[84,154,121],[33,10333,10335,10338],{"style":10334},"top:-2.971em;",[33,10336],{"className":10337,"style":7526},[113],[33,10339,10341],{"className":10340},[118,7530,1076,121],[33,10342,10344,10347],{"className":10343},[84,121],[33,10345,402],{"className":10346},[84,121],[33,10348,2195],{"className":10349},[84,154,121],[33,10351,160],{"className":10352},[159],[33,10354,10356],{"className":10355},[101],[33,10357,10360],{"className":10358,"style":10359},[105],"height:0.344em;",[33,10361],{},[33,10363,10365],{"className":10364,"style":10297},[173,118,7530,10296,121,1070],[33,10366,10368],{"className":10367},[1075,1076,121],[33,10369,1035],{"className":10370},[121],[33,10372,160],{"className":10373},[159],[33,10375,10377],{"className":10376},[101],[33,10378,10380],{"className":10379,"style":3734},[105],[33,10381],{},[33,10383],{"className":10384},[173,89],[15,10386,10387],{},"Y la serie queda:",[15,10389,10391],{"className":10390},[590],[33,10392,10394,10530],{"className":10393},[36],[33,10395,10397],{"className":10396},[40],[42,10398,10399],{"xmlns":44},[46,10400,10401,10527],{},[49,10402,10403,10405,10407,10421,10467,10469,10475,10489],{},[604,10404,7297],{},[274,10406,276],{},[7301,10408,10409,10411,10419],{},[274,10410,7305],{},[49,10412,10413,10415,10417],{},[55,10414,2195],{},[274,10416,276],{},[59,10418,515],{},[55,10420,7317],{"mathvariant":7316},[52,10422,10423,10439],{},[49,10424,10425,10427,10437],{},[274,10426,1026],{"fence":71},[52,10428,10429,10435],{"linethickness":4319},[49,10430,10431,10433],{},[59,10432,402],{},[55,10434,2195],{},[55,10436,2195],{},[274,10438,1035],{"fence":71},[49,10440,10441,10455,10457,10459,10461,10463,10465],{},[1020,10442,10443,10445],{},[59,10444,402],{},[49,10446,10447,10449,10451,10453],{},[59,10448,402],{},[55,10450,2195],{},[274,10452,1314],{},[59,10454,270],{},[274,10456,1026],{"stretchy":9814},[59,10458,402],{},[55,10460,2195],{},[274,10462,1314],{},[59,10464,270],{},[274,10466,1035],{"stretchy":9814},[274,10468,276],{},[52,10470,10471,10473],{},[59,10472,270],{},[59,10474,402],{},[7301,10476,10477,10479,10487],{},[274,10478,7305],{},[49,10480,10481,10483,10485],{},[55,10482,2195],{},[274,10484,276],{},[59,10486,515],{},[55,10488,7317],{"mathvariant":7316},[52,10490,10491,10507],{},[49,10492,10493,10495,10505],{},[274,10494,1026],{"fence":71},[52,10496,10497,10503],{"linethickness":4319},[49,10498,10499,10501],{},[59,10500,402],{},[55,10502,2195],{},[55,10504,2195],{},[274,10506,1035],{"fence":71},[49,10508,10509,10515,10517,10519,10521,10523,10525],{},[1020,10510,10511,10513],{},[59,10512,61],{},[55,10514,2195],{},[274,10516,1026],{"stretchy":9814},[59,10518,402],{},[55,10520,2195],{},[274,10522,1314],{},[59,10524,270],{},[274,10526,1035],{"stretchy":9814},[63,10528,10529],{"encoding":65},"\\text{Media} = \\sum_{n=0}^{\\infty} \\frac{\\binom{2n}{n}}{2^{2n+1}(2n+1)} = \\frac{1}{2} \\sum_{n=0}^{\\infty} \\frac{\\binom{2n}{n}}{4^n(2n+1)}",[33,10531,10533,10554,10833],{"className":10532,"ariaHidden":71},[70],[33,10534,10536,10539,10545,10548,10551],{"className":10535},[75],[33,10537],{"className":10538,"style":7024},[79],[33,10540,10542],{"className":10541},[84,623],[33,10543,7297],{"className":10544},[84],[33,10546],{"className":10547,"style":365},[364],[33,10549,276],{"className":10550},[369],[33,10552],{"className":10553,"style":365},[364],[33,10555,10557,10561,10624,10627,10824,10827,10830],{"className":10556},[75],[33,10558],{"className":10559,"style":10560},[79],"height:2.03em;vertical-align:-0.52em;",[33,10562,10564,10567],{"className":10563},[7400],[33,10565,7305],{"className":10566,"style":7406},[7400,7404,7405],[33,10568,10570],{"className":10569},[1154],[33,10571,10573,10616],{"className":10572},[96,97],[33,10574,10576,10613],{"className":10575},[101],[33,10577,10579,10599],{"className":10578,"style":7419},[105],[33,10580,10581,10584],{"style":7422},[33,10582],{"className":10583,"style":1171},[113],[33,10585,10587],{"className":10586},[118,119,120,121],[33,10588,10590,10593,10596],{"className":10589},[84,121],[33,10591,2195],{"className":10592},[84,154,121],[33,10594,276],{"className":10595},[369,121],[33,10597,515],{"className":10598},[84,121],[33,10600,10601,10604],{"style":7443},[33,10602],{"className":10603,"style":1171},[113],[33,10605,10607],{"className":10606},[118,119,120,121],[33,10608,10610],{"className":10609},[84,121],[33,10611,7317],{"className":10612},[84,121],[33,10614,160],{"className":10615},[159],[33,10617,10619],{"className":10618},[101],[33,10620,10622],{"className":10621,"style":7465},[105],[33,10623],{},[33,10625],{"className":10626,"style":1450},[364],[33,10628,10630,10633,10821],{"className":10629},[84],[33,10631],{"className":10632},[88,89],[33,10634,10636],{"className":10635},[52],[33,10637,10639,10812],{"className":10638},[96,97],[33,10640,10642,10809],{"className":10641},[101],[33,10643,10645,10715,10723],{"className":10644,"style":10246},[105],[33,10646,10647,10650],{"style":109},[33,10648],{"className":10649,"style":114},[113],[33,10651,10653],{"className":10652},[118,119,120,121],[33,10654,10656,10697,10700,10703,10706,10709,10712],{"className":10655},[84,121],[33,10657,10659,10662],{"className":10658},[84,121],[33,10660,402],{"className":10661},[84,121],[33,10663,10665],{"className":10664},[1154],[33,10666,10668],{"className":10667},[96],[33,10669,10671],{"className":10670},[101],[33,10672,10674],{"className":10673,"style":7519},[105],[33,10675,10676,10679],{"style":7522},[33,10677],{"className":10678,"style":7526},[113],[33,10680,10682],{"className":10681},[118,7530,1076,121],[33,10683,10685,10688,10691,10694],{"className":10684},[84,121],[33,10686,402],{"className":10687},[84,121],[33,10689,2195],{"className":10690},[84,154,121],[33,10692,1314],{"className":10693},[2225,121],[33,10695,270],{"className":10696},[84,121],[33,10698,1026],{"className":10699},[88,121],[33,10701,402],{"className":10702},[84,121],[33,10704,2195],{"className":10705},[84,154,121],[33,10707,1314],{"className":10708},[2225,121],[33,10710,270],{"className":10711},[84,121],[33,10713,1035],{"className":10714},[173,121],[33,10716,10717,10720],{"style":130},[33,10718],{"className":10719,"style":114},[113],[33,10721],{"className":10722,"style":138},[137],[33,10724,10725,10728],{"style":10280},[33,10726],{"className":10727,"style":114},[113],[33,10729,10731],{"className":10730},[118,119,120,121],[33,10732,10734],{"className":10733},[84,121],[33,10735,10737,10746,10800],{"className":10736},[84,121],[33,10738,10740],{"className":10739,"style":10297},[88,118,7530,10296,121,1070],[33,10741,10743],{"className":10742},[1075,1076,121],[33,10744,1026],{"className":10745},[121],[33,10747,10749],{"className":10748},[52],[33,10750,10752,10792],{"className":10751},[96,97],[33,10753,10755,10789],{"className":10754},[101],[33,10756,10758,10772],{"className":10757,"style":10316},[105],[33,10759,10760,10763],{"style":10319},[33,10761],{"className":10762,"style":7526},[113],[33,10764,10766],{"className":10765},[118,7530,1076,121],[33,10767,10769],{"className":10768},[84,121],[33,10770,2195],{"className":10771},[84,154,121],[33,10773,10774,10777],{"style":10334},[33,10775],{"className":10776,"style":7526},[113],[33,10778,10780],{"className":10779},[118,7530,1076,121],[33,10781,10783,10786],{"className":10782},[84,121],[33,10784,402],{"className":10785},[84,121],[33,10787,2195],{"className":10788},[84,154,121],[33,10790,160],{"className":10791},[159],[33,10793,10795],{"className":10794},[101],[33,10796,10798],{"className":10797,"style":10359},[105],[33,10799],{},[33,10801,10803],{"className":10802,"style":10297},[173,118,7530,10296,121,1070],[33,10804,10806],{"className":10805},[1075,1076,121],[33,10807,1035],{"className":10808},[121],[33,10810,160],{"className":10811},[159],[33,10813,10815],{"className":10814},[101],[33,10816,10819],{"className":10817,"style":10818},[105],"height:0.52em;",[33,10820],{},[33,10822],{"className":10823},[173,89],[33,10825],{"className":10826,"style":365},[364],[33,10828,276],{"className":10829},[369],[33,10831],{"className":10832,"style":365},[364],[33,10834,10836,10839,10907,10910,10973,10976],{"className":10835},[75],[33,10837],{"className":10838,"style":10560},[79],[33,10840,10842,10845,10904],{"className":10841},[84],[33,10843],{"className":10844},[88,89],[33,10846,10848],{"className":10847},[52],[33,10849,10851,10896],{"className":10850},[96,97],[33,10852,10854,10893],{"className":10853},[101],[33,10855,10857,10871,10879],{"className":10856,"style":310},[105],[33,10858,10859,10862],{"style":109},[33,10860],{"className":10861,"style":114},[113],[33,10863,10865],{"className":10864},[118,119,120,121],[33,10866,10868],{"className":10867},[84,121],[33,10869,402],{"className":10870},[84,121],[33,10872,10873,10876],{"style":130},[33,10874],{"className":10875,"style":114},[113],[33,10877],{"className":10878,"style":138},[137],[33,10880,10881,10884],{"style":141},[33,10882],{"className":10883,"style":114},[113],[33,10885,10887],{"className":10886},[118,119,120,121],[33,10888,10890],{"className":10889},[84,121],[33,10891,270],{"className":10892},[84,121],[33,10894,160],{"className":10895},[159],[33,10897,10899],{"className":10898},[101],[33,10900,10902],{"className":10901,"style":167},[105],[33,10903],{},[33,10905],{"className":10906},[173,89],[33,10908],{"className":10909,"style":1450},[364],[33,10911,10913,10916],{"className":10912},[7400],[33,10914,7305],{"className":10915,"style":7406},[7400,7404,7405],[33,10917,10919],{"className":10918},[1154],[33,10920,10922,10965],{"className":10921},[96,97],[33,10923,10925,10962],{"className":10924},[101],[33,10926,10928,10948],{"className":10927,"style":7419},[105],[33,10929,10930,10933],{"style":7422},[33,10931],{"className":10932,"style":1171},[113],[33,10934,10936],{"className":10935},[118,119,120,121],[33,10937,10939,10942,10945],{"className":10938},[84,121],[33,10940,2195],{"className":10941},[84,154,121],[33,10943,276],{"className":10944},[369,121],[33,10946,515],{"className":10947},[84,121],[33,10949,10950,10953],{"style":7443},[33,10951],{"className":10952,"style":1171},[113],[33,10954,10956],{"className":10955},[118,119,120,121],[33,10957,10959],{"className":10958},[84,121],[33,10960,7317],{"className":10961},[84,121],[33,10963,160],{"className":10964},[159],[33,10966,10968],{"className":10967},[101],[33,10969,10971],{"className":10970,"style":7465},[105],[33,10972],{},[33,10974],{"className":10975,"style":1450},[364],[33,10977,10979,10982,11158],{"className":10978},[84],[33,10980],{"className":10981},[88,89],[33,10983,10985],{"className":10984},[52],[33,10986,10988,11150],{"className":10987},[96,97],[33,10989,10991,11147],{"className":10990},[101],[33,10992,10994,11053,11061],{"className":10993,"style":10246},[105],[33,10995,10996,10999],{"style":109},[33,10997],{"className":10998,"style":114},[113],[33,11000,11002],{"className":11001},[118,119,120,121],[33,11003,11005,11035,11038,11041,11044,11047,11050],{"className":11004},[84,121],[33,11006,11008,11011],{"className":11007},[84,121],[33,11009,61],{"className":11010},[84,121],[33,11012,11014],{"className":11013},[1154],[33,11015,11017],{"className":11016},[96],[33,11018,11020],{"className":11019},[101],[33,11021,11024],{"className":11022,"style":11023},[105],"height:0.5935em;",[33,11025,11026,11029],{"style":7522},[33,11027],{"className":11028,"style":7526},[113],[33,11030,11032],{"className":11031},[118,7530,1076,121],[33,11033,2195],{"className":11034},[84,154,121],[33,11036,1026],{"className":11037},[88,121],[33,11039,402],{"className":11040},[84,121],[33,11042,2195],{"className":11043},[84,154,121],[33,11045,1314],{"className":11046},[2225,121],[33,11048,270],{"className":11049},[84,121],[33,11051,1035],{"className":11052},[173,121],[33,11054,11055,11058],{"style":130},[33,11056],{"className":11057,"style":114},[113],[33,11059],{"className":11060,"style":138},[137],[33,11062,11063,11066],{"style":10280},[33,11064],{"className":11065,"style":114},[113],[33,11067,11069],{"className":11068},[118,119,120,121],[33,11070,11072],{"className":11071},[84,121],[33,11073,11075,11084,11138],{"className":11074},[84,121],[33,11076,11078],{"className":11077,"style":10297},[88,118,7530,10296,121,1070],[33,11079,11081],{"className":11080},[1075,1076,121],[33,11082,1026],{"className":11083},[121],[33,11085,11087],{"className":11086},[52],[33,11088,11090,11130],{"className":11089},[96,97],[33,11091,11093,11127],{"className":11092},[101],[33,11094,11096,11110],{"className":11095,"style":10316},[105],[33,11097,11098,11101],{"style":10319},[33,11099],{"className":11100,"style":7526},[113],[33,11102,11104],{"className":11103},[118,7530,1076,121],[33,11105,11107],{"className":11106},[84,121],[33,11108,2195],{"className":11109},[84,154,121],[33,11111,11112,11115],{"style":10334},[33,11113],{"className":11114,"style":7526},[113],[33,11116,11118],{"className":11117},[118,7530,1076,121],[33,11119,11121,11124],{"className":11120},[84,121],[33,11122,402],{"className":11123},[84,121],[33,11125,2195],{"className":11126},[84,154,121],[33,11128,160],{"className":11129},[159],[33,11131,11133],{"className":11132},[101],[33,11134,11136],{"className":11135,"style":10359},[105],[33,11137],{},[33,11139,11141],{"className":11140,"style":10297},[173,118,7530,10296,121,1070],[33,11142,11144],{"className":11143},[1075,1076,121],[33,11145,1035],{"className":11146},[121],[33,11148,160],{"className":11149},[159],[33,11151,11153],{"className":11152},[101],[33,11154,11156],{"className":11155,"style":10818},[105],[33,11157],{},[33,11159],{"className":11160},[173,89],[15,11162,11163,11164,11167],{},"Ahora viene el momento que a Matt Parker le hace «enfadar de pura incredulidad». Resulta que existe una función clásica del análisis matemático llamada ",[237,11165,11166],{},"arcoseno"," — la función inversa del seno — cuya expansión en serie de potencias es:",[15,11169,11171],{"className":11170},[590],[33,11172,11174,11267],{"className":11173},[36],[33,11175,11177],{"className":11176},[40],[42,11178,11179],{"xmlns":44},[46,11180,11181,11264],{},[49,11182,11183,11186,11189,11191,11194,11196,11198,11212,11236,11238],{},[55,11184,11185],{},"arcsin",[274,11187,11188],{},"⁡",[274,11190,1026],{"stretchy":9814},[55,11192,11193],{},"x",[274,11195,1035],{"stretchy":9814},[274,11197,276],{},[7301,11199,11200,11202,11210],{},[274,11201,7305],{},[49,11203,11204,11206,11208],{},[55,11205,2195],{},[274,11207,276],{},[59,11209,515],{},[55,11211,7317],{"mathvariant":7316},[52,11213,11214,11230],{},[49,11215,11216,11218,11228],{},[274,11217,1026],{"fence":71},[52,11219,11220,11226],{"linethickness":4319},[49,11221,11222,11224],{},[59,11223,402],{},[55,11225,2195],{},[55,11227,2195],{},[274,11229,1035],{"fence":71},[1020,11231,11232,11234],{},[59,11233,61],{},[55,11235,2195],{},[274,11237,7342],{},[52,11239,11240,11254],{},[1020,11241,11242,11244],{},[55,11243,11193],{},[49,11245,11246,11248,11250,11252],{},[59,11247,402],{},[55,11249,2195],{},[274,11251,1314],{},[59,11253,270],{},[49,11255,11256,11258,11260,11262],{},[59,11257,402],{},[55,11259,2195],{},[274,11261,1314],{},[59,11263,270],{},[63,11265,11266],{"encoding":65},"\\arcsin(x) = \\sum_{n=0}^{\\infty} \\frac{\\binom{2n}{n}}{4^n} \\cdot \\frac{x^{2n+1}}{2n+1}",[33,11268,11270,11297,11545],{"className":11269,"ariaHidden":71},[70],[33,11271,11273,11276,11279,11282,11285,11288,11291,11294],{"className":11272},[75],[33,11274],{"className":11275,"style":9835},[79],[33,11277,11185],{"className":11278},[7400],[33,11280,1026],{"className":11281},[88],[33,11283,11193],{"className":11284},[84,154],[33,11286,1035],{"className":11287},[173],[33,11289],{"className":11290,"style":365},[364],[33,11292,276],{"className":11293},[369],[33,11295],{"className":11296,"style":365},[364],[33,11298,11300,11304,11367,11370,11536,11539,11542],{"className":11299},[75],[33,11301],{"className":11302,"style":11303},[79],"height:1.855em;vertical-align:-0.345em;",[33,11305,11307,11310],{"className":11306},[7400],[33,11308,7305],{"className":11309,"style":7406},[7400,7404,7405],[33,11311,11313],{"className":11312},[1154],[33,11314,11316,11359],{"className":11315},[96,97],[33,11317,11319,11356],{"className":11318},[101],[33,11320,11322,11342],{"className":11321,"style":7419},[105],[33,11323,11324,11327],{"style":7422},[33,11325],{"className":11326,"style":1171},[113],[33,11328,11330],{"className":11329},[118,119,120,121],[33,11331,11333,11336,11339],{"className":11332},[84,121],[33,11334,2195],{"className":11335},[84,154,121],[33,11337,276],{"className":11338},[369,121],[33,11340,515],{"className":11341},[84,121],[33,11343,11344,11347],{"style":7443},[33,11345],{"className":11346,"style":1171},[113],[33,11348,11350],{"className":11349},[118,119,120,121],[33,11351,11353],{"className":11352},[84,121],[33,11354,7317],{"className":11355},[84,121],[33,11357,160],{"className":11358},[159],[33,11360,11362],{"className":11361},[101],[33,11363,11365],{"className":11364,"style":7465},[105],[33,11366],{},[33,11368],{"className":11369,"style":1450},[364],[33,11371,11373,11376,11533],{"className":11372},[84],[33,11374],{"className":11375},[88,89],[33,11377,11379],{"className":11378},[52],[33,11380,11382,11525],{"className":11381},[96,97],[33,11383,11385,11522],{"className":11384},[101],[33,11386,11388,11428,11436],{"className":11387,"style":10246},[105],[33,11389,11390,11393],{"style":109},[33,11391],{"className":11392,"style":114},[113],[33,11394,11396],{"className":11395},[118,119,120,121],[33,11397,11399],{"className":11398},[84,121],[33,11400,11402,11405],{"className":11401},[84,121],[33,11403,61],{"className":11404},[84,121],[33,11406,11408],{"className":11407},[1154],[33,11409,11411],{"className":11410},[96],[33,11412,11414],{"className":11413},[101],[33,11415,11417],{"className":11416,"style":11023},[105],[33,11418,11419,11422],{"style":7522},[33,11420],{"className":11421,"style":7526},[113],[33,11423,11425],{"className":11424},[118,7530,1076,121],[33,11426,2195],{"className":11427},[84,154,121],[33,11429,11430,11433],{"style":130},[33,11431],{"className":11432,"style":114},[113],[33,11434],{"className":11435,"style":138},[137],[33,11437,11438,11441],{"style":10280},[33,11439],{"className":11440,"style":114},[113],[33,11442,11444],{"className":11443},[118,119,120,121],[33,11445,11447],{"className":11446},[84,121],[33,11448,11450,11459,11513],{"className":11449},[84,121],[33,11451,11453],{"className":11452,"style":10297},[88,118,7530,10296,121,1070],[33,11454,11456],{"className":11455},[1075,1076,121],[33,11457,1026],{"className":11458},[121],[33,11460,11462],{"className":11461},[52],[33,11463,11465,11505],{"className":11464},[96,97],[33,11466,11468,11502],{"className":11467},[101],[33,11469,11471,11485],{"className":11470,"style":10316},[105],[33,11472,11473,11476],{"style":10319},[33,11474],{"className":11475,"style":7526},[113],[33,11477,11479],{"className":11478},[118,7530,1076,121],[33,11480,11482],{"className":11481},[84,121],[33,11483,2195],{"className":11484},[84,154,121],[33,11486,11487,11490],{"style":10334},[33,11488],{"className":11489,"style":7526},[113],[33,11491,11493],{"className":11492},[118,7530,1076,121],[33,11494,11496,11499],{"className":11495},[84,121],[33,11497,402],{"className":11498},[84,121],[33,11500,2195],{"className":11501},[84,154,121],[33,11503,160],{"className":11504},[159],[33,11506,11508],{"className":11507},[101],[33,11509,11511],{"className":11510,"style":10359},[105],[33,11512],{},[33,11514,11516],{"className":11515,"style":10297},[173,118,7530,10296,121,1070],[33,11517,11519],{"className":11518},[1075,1076,121],[33,11520,1035],{"className":11521},[121],[33,11523,160],{"className":11524},[159],[33,11526,11528],{"className":11527},[101],[33,11529,11531],{"className":11530,"style":167},[105],[33,11532],{},[33,11534],{"className":11535},[173,89],[33,11537],{"className":11538,"style":2221},[364],[33,11540,7342],{"className":11541},[2225],[33,11543],{"className":11544,"style":2221},[364],[33,11546,11548,11552],{"className":11547},[75],[33,11549],{"className":11550,"style":11551},[79],"height:1.4213em;vertical-align:-0.4033em;",[33,11553,11555,11558,11667],{"className":11554},[84],[33,11556],{"className":11557},[88,89],[33,11559,11561],{"className":11560},[52],[33,11562,11564,11659],{"className":11563},[96,97],[33,11565,11567,11656],{"className":11566},[101],[33,11568,11571,11594,11602],{"className":11569,"style":11570},[105],"height:1.0179em;",[33,11572,11573,11576],{"style":109},[33,11574],{"className":11575,"style":114},[113],[33,11577,11579],{"className":11578},[118,119,120,121],[33,11580,11582,11585,11588,11591],{"className":11581},[84,121],[33,11583,402],{"className":11584},[84,121],[33,11586,2195],{"className":11587},[84,154,121],[33,11589,1314],{"className":11590},[2225,121],[33,11592,270],{"className":11593},[84,121],[33,11595,11596,11599],{"style":130},[33,11597],{"className":11598,"style":114},[113],[33,11600],{"className":11601,"style":138},[137],[33,11603,11604,11607],{"style":141},[33,11605],{"className":11606,"style":114},[113],[33,11608,11610],{"className":11609},[118,119,120,121],[33,11611,11613],{"className":11612},[84,121],[33,11614,11616,11619],{"className":11615},[84,121],[33,11617,11193],{"className":11618},[84,154,121],[33,11620,11622],{"className":11621},[1154],[33,11623,11625],{"className":11624},[96],[33,11626,11628],{"className":11627},[101],[33,11629,11632],{"className":11630,"style":11631},[105],"height:0.8913em;",[33,11633,11635,11638],{"style":11634},"top:-2.931em;margin-right:0.0714em;",[33,11636],{"className":11637,"style":7526},[113],[33,11639,11641],{"className":11640},[118,7530,1076,121],[33,11642,11644,11647,11650,11653],{"className":11643},[84,121],[33,11645,402],{"className":11646},[84,121],[33,11648,2195],{"className":11649},[84,154,121],[33,11651,1314],{"className":11652},[2225,121],[33,11654,270],{"className":11655},[84,121],[33,11657,160],{"className":11658},[159],[33,11660,11662],{"className":11661},[101],[33,11663,11665],{"className":11664,"style":3734},[105],[33,11666],{},[33,11668],{"className":11669},[173,89],[15,11671,11672,11673,11701,11702,11730,11731,174],{},"¿Te parece familiar? Es exactamente nuestra serie, multiplicada por ",[33,11674,11676,11689],{"className":11675},[36],[33,11677,11679],{"className":11678},[40],[42,11680,11681],{"xmlns":44},[46,11682,11683,11687],{},[49,11684,11685],{},[59,11686,402],{},[63,11688,402],{"encoding":65},[33,11690,11692],{"className":11691,"ariaHidden":71},[70],[33,11693,11695,11698],{"className":11694},[75],[33,11696],{"className":11697,"style":379},[79],[33,11699,402],{"className":11700},[84]," y con ",[33,11703,11705,11718],{"className":11704},[36],[33,11706,11708],{"className":11707},[40],[42,11709,11710],{"xmlns":44},[46,11711,11712,11716],{},[49,11713,11714],{},[55,11715,11193],{},[63,11717,11193],{"encoding":65},[33,11719,11721],{"className":11720,"ariaHidden":71},[70],[33,11722,11724,11727],{"className":11723},[75],[33,11725],{"className":11726,"style":204},[79],[33,11728,11193],{"className":11729},[84,154]," en lugar de ",[33,11732,11734,11747],{"className":11733},[36],[33,11735,11737],{"className":11736},[40],[42,11738,11739],{"xmlns":44},[46,11740,11741,11745],{},[49,11742,11743],{},[59,11744,270],{},[63,11746,270],{"encoding":65},[33,11748,11750],{"className":11749,"ariaHidden":71},[70],[33,11751,11753,11756],{"className":11752},[75],[33,11754],{"className":11755,"style":379},[79],[33,11757,270],{"className":11758},[84],[15,11760,11761,11762,11813],{},"Si evaluamos en ",[33,11763,11765,11783],{"className":11764},[36],[33,11766,11768],{"className":11767},[40],[42,11769,11770],{"xmlns":44},[46,11771,11772,11780],{},[49,11773,11774,11776,11778],{},[55,11775,11193],{},[274,11777,276],{},[59,11779,270],{},[63,11781,11782],{"encoding":65},"x = 1",[33,11784,11786,11804],{"className":11785,"ariaHidden":71},[70],[33,11787,11789,11792,11795,11798,11801],{"className":11788},[75],[33,11790],{"className":11791,"style":204},[79],[33,11793,11193],{"className":11794},[84,154],[33,11796],{"className":11797,"style":365},[364],[33,11799,276],{"className":11800},[369],[33,11802],{"className":11803,"style":365},[364],[33,11805,11807,11810],{"className":11806},[75],[33,11808],{"className":11809,"style":379},[79],[33,11811,270],{"className":11812},[84],":",[15,11815,11817],{"className":11816},[590],[33,11818,11820,11896],{"className":11819},[36],[33,11821,11823],{"className":11822},[40],[42,11824,11825],{"xmlns":44},[46,11826,11827,11893],{},[49,11828,11829,11831,11833,11835,11837,11839,11841,11855],{},[55,11830,11185],{},[274,11832,11188],{},[274,11834,1026],{"stretchy":9814},[59,11836,270],{},[274,11838,1035],{"stretchy":9814},[274,11840,276],{},[7301,11842,11843,11845,11853],{},[274,11844,7305],{},[49,11846,11847,11849,11851],{},[55,11848,2195],{},[274,11850,276],{},[59,11852,515],{},[55,11854,7317],{"mathvariant":7316},[52,11856,11857,11873],{},[49,11858,11859,11861,11871],{},[274,11860,1026],{"fence":71},[52,11862,11863,11869],{"linethickness":4319},[49,11864,11865,11867],{},[59,11866,402],{},[55,11868,2195],{},[55,11870,2195],{},[274,11872,1035],{"fence":71},[49,11874,11875,11881,11883,11885,11887,11889,11891],{},[1020,11876,11877,11879],{},[59,11878,61],{},[55,11880,2195],{},[274,11882,1026],{"stretchy":9814},[59,11884,402],{},[55,11886,2195],{},[274,11888,1314],{},[59,11890,270],{},[274,11892,1035],{"stretchy":9814},[63,11894,11895],{"encoding":65},"\\arcsin(1) = \\sum_{n=0}^{\\infty} \\frac{\\binom{2n}{n}}{4^n(2n+1)}",[33,11897,11899,11926],{"className":11898,"ariaHidden":71},[70],[33,11900,11902,11905,11908,11911,11914,11917,11920,11923],{"className":11901},[75],[33,11903],{"className":11904,"style":9835},[79],[33,11906,11185],{"className":11907},[7400],[33,11909,1026],{"className":11910},[88],[33,11912,270],{"className":11913},[84],[33,11915,1035],{"className":11916},[173],[33,11918],{"className":11919,"style":365},[364],[33,11921,276],{"className":11922},[369],[33,11924],{"className":11925,"style":365},[364],[33,11927,11929,11932,11995,11998],{"className":11928},[75],[33,11930],{"className":11931,"style":10560},[79],[33,11933,11935,11938],{"className":11934},[7400],[33,11936,7305],{"className":11937,"style":7406},[7400,7404,7405],[33,11939,11941],{"className":11940},[1154],[33,11942,11944,11987],{"className":11943},[96,97],[33,11945,11947,11984],{"className":11946},[101],[33,11948,11950,11970],{"className":11949,"style":7419},[105],[33,11951,11952,11955],{"style":7422},[33,11953],{"className":11954,"style":1171},[113],[33,11956,11958],{"className":11957},[118,119,120,121],[33,11959,11961,11964,11967],{"className":11960},[84,121],[33,11962,2195],{"className":11963},[84,154,121],[33,11965,276],{"className":11966},[369,121],[33,11968,515],{"className":11969},[84,121],[33,11971,11972,11975],{"style":7443},[33,11973],{"className":11974,"style":1171},[113],[33,11976,11978],{"className":11977},[118,119,120,121],[33,11979,11981],{"className":11980},[84,121],[33,11982,7317],{"className":11983},[84,121],[33,11985,160],{"className":11986},[159],[33,11988,11990],{"className":11989},[101],[33,11991,11993],{"className":11992,"style":7465},[105],[33,11994],{},[33,11996],{"className":11997,"style":1450},[364],[33,11999,12001,12004,12179],{"className":12000},[84],[33,12002],{"className":12003},[88,89],[33,12005,12007],{"className":12006},[52],[33,12008,12010,12171],{"className":12009},[96,97],[33,12011,12013,12168],{"className":12012},[101],[33,12014,12016,12074,12082],{"className":12015,"style":10246},[105],[33,12017,12018,12021],{"style":109},[33,12019],{"className":12020,"style":114},[113],[33,12022,12024],{"className":12023},[118,119,120,121],[33,12025,12027,12056,12059,12062,12065,12068,12071],{"className":12026},[84,121],[33,12028,12030,12033],{"className":12029},[84,121],[33,12031,61],{"className":12032},[84,121],[33,12034,12036],{"className":12035},[1154],[33,12037,12039],{"className":12038},[96],[33,12040,12042],{"className":12041},[101],[33,12043,12045],{"className":12044,"style":11023},[105],[33,12046,12047,12050],{"style":7522},[33,12048],{"className":12049,"style":7526},[113],[33,12051,12053],{"className":12052},[118,7530,1076,121],[33,12054,2195],{"className":12055},[84,154,121],[33,12057,1026],{"className":12058},[88,121],[33,12060,402],{"className":12061},[84,121],[33,12063,2195],{"className":12064},[84,154,121],[33,12066,1314],{"className":12067},[2225,121],[33,12069,270],{"className":12070},[84,121],[33,12072,1035],{"className":12073},[173,121],[33,12075,12076,12079],{"style":130},[33,12077],{"className":12078,"style":114},[113],[33,12080],{"className":12081,"style":138},[137],[33,12083,12084,12087],{"style":10280},[33,12085],{"className":12086,"style":114},[113],[33,12088,12090],{"className":12089},[118,119,120,121],[33,12091,12093],{"className":12092},[84,121],[33,12094,12096,12105,12159],{"className":12095},[84,121],[33,12097,12099],{"className":12098,"style":10297},[88,118,7530,10296,121,1070],[33,12100,12102],{"className":12101},[1075,1076,121],[33,12103,1026],{"className":12104},[121],[33,12106,12108],{"className":12107},[52],[33,12109,12111,12151],{"className":12110},[96,97],[33,12112,12114,12148],{"className":12113},[101],[33,12115,12117,12131],{"className":12116,"style":10316},[105],[33,12118,12119,12122],{"style":10319},[33,12120],{"className":12121,"style":7526},[113],[33,12123,12125],{"className":12124},[118,7530,1076,121],[33,12126,12128],{"className":12127},[84,121],[33,12129,2195],{"className":12130},[84,154,121],[33,12132,12133,12136],{"style":10334},[33,12134],{"className":12135,"style":7526},[113],[33,12137,12139],{"className":12138},[118,7530,1076,121],[33,12140,12142,12145],{"className":12141},[84,121],[33,12143,402],{"className":12144},[84,121],[33,12146,2195],{"className":12147},[84,154,121],[33,12149,160],{"className":12150},[159],[33,12152,12154],{"className":12153},[101],[33,12155,12157],{"className":12156,"style":10359},[105],[33,12158],{},[33,12160,12162],{"className":12161,"style":10297},[173,118,7530,10296,121,1070],[33,12163,12165],{"className":12164},[1075,1076,121],[33,12166,1035],{"className":12167},[121],[33,12169,160],{"className":12170},[159],[33,12172,12174],{"className":12173},[101],[33,12175,12177],{"className":12176,"style":10818},[105],[33,12178],{},[33,12180],{"className":12181},[173,89],[15,12183,12184,12185,12231,12232,12260,12261,11813],{},"¿Y cuánto vale ",[33,12186,12188,12210],{"className":12187},[36],[33,12189,12191],{"className":12190},[40],[42,12192,12193],{"xmlns":44},[46,12194,12195,12207],{},[49,12196,12197,12199,12201,12203,12205],{},[55,12198,11185],{},[274,12200,11188],{},[274,12202,1026],{"stretchy":9814},[59,12204,270],{},[274,12206,1035],{"stretchy":9814},[63,12208,12209],{"encoding":65},"\\arcsin(1)",[33,12211,12213],{"className":12212,"ariaHidden":71},[70],[33,12214,12216,12219,12222,12225,12228],{"className":12215},[75],[33,12217],{"className":12218,"style":9835},[79],[33,12220,11185],{"className":12221},[7400],[33,12223,1026],{"className":12224},[88],[33,12226,270],{"className":12227},[84],[33,12229,1035],{"className":12230},[173],"? Es el ángulo cuyo seno es ",[33,12233,12235,12248],{"className":12234},[36],[33,12236,12238],{"className":12237},[40],[42,12239,12240],{"xmlns":44},[46,12241,12242,12246],{},[49,12243,12244],{},[59,12245,270],{},[63,12247,270],{"encoding":65},[33,12249,12251],{"className":12250,"ariaHidden":71},[70],[33,12252,12254,12257],{"className":12253},[75],[33,12255],{"className":12256,"style":379},[79],[33,12258,270],{"className":12259},[84],", es decir, ",[33,12262,12264,12282],{"className":12263},[36],[33,12265,12267],{"className":12266},[40],[42,12268,12269],{"xmlns":44},[46,12270,12271,12279],{},[49,12272,12273,12276],{},[59,12274,12275],{},"90",[55,12277,12278],{"mathvariant":7316},"°",[63,12280,12281],{"encoding":65},"90°",[33,12283,12285],{"className":12284,"ariaHidden":71},[70],[33,12286,12288,12291],{"className":12287},[75],[33,12289],{"className":12290,"style":7024},[79],[33,12292,12281],{"className":12293},[84],[15,12295,12297],{"className":12296},[590],[33,12298,12300,12330],{"className":12299},[36],[33,12301,12303],{"className":12302},[40],[42,12304,12305],{"xmlns":44},[46,12306,12307,12327],{},[49,12308,12309,12311,12313,12315,12317,12319,12321],{},[55,12310,11185],{},[274,12312,11188],{},[274,12314,1026],{"stretchy":9814},[59,12316,270],{},[274,12318,1035],{"stretchy":9814},[274,12320,276],{},[52,12322,12323,12325],{},[55,12324,57],{},[59,12326,402],{},[63,12328,12329],{"encoding":65},"\\arcsin(1) = \\frac{\\pi}{2}",[33,12331,12333,12360],{"className":12332,"ariaHidden":71},[70],[33,12334,12336,12339,12342,12345,12348,12351,12354,12357],{"className":12335},[75],[33,12337],{"className":12338,"style":9835},[79],[33,12340,11185],{"className":12341},[7400],[33,12343,1026],{"className":12344},[88],[33,12346,270],{"className":12347},[84],[33,12349,1035],{"className":12350},[173],[33,12352],{"className":12353,"style":365},[364],[33,12355,276],{"className":12356},[369],[33,12358],{"className":12359,"style":365},[364],[33,12361,12363,12366],{"className":12362},[75],[33,12364],{"className":12365,"style":80},[79],[33,12367,12369,12372,12431],{"className":12368},[84],[33,12370],{"className":12371},[88,89],[33,12373,12375],{"className":12374},[52],[33,12376,12378,12423],{"className":12377},[96,97],[33,12379,12381,12420],{"className":12380},[101],[33,12382,12384,12398,12406],{"className":12383,"style":106},[105],[33,12385,12386,12389],{"style":109},[33,12387],{"className":12388,"style":114},[113],[33,12390,12392],{"className":12391},[118,119,120,121],[33,12393,12395],{"className":12394},[84,121],[33,12396,402],{"className":12397},[84,121],[33,12399,12400,12403],{"style":130},[33,12401],{"className":12402,"style":114},[113],[33,12404],{"className":12405,"style":138},[137],[33,12407,12408,12411],{"style":141},[33,12409],{"className":12410,"style":114},[113],[33,12412,12414],{"className":12413},[118,119,120,121],[33,12415,12417],{"className":12416},[84,121],[33,12418,57],{"className":12419,"style":155},[84,154,121],[33,12421,160],{"className":12422},[159],[33,12424,12426],{"className":12425},[101],[33,12427,12429],{"className":12428,"style":167},[105],[33,12430],{},[33,12432],{"className":12433},[173,89],[15,12435,12436],{},"Por lo tanto:",[15,12438,12440],{"className":12439},[590],[33,12441,12443,12481],{"className":12442},[36],[33,12444,12446],{"className":12445},[40],[42,12447,12448],{"xmlns":44},[46,12449,12450,12478],{},[49,12451,12452,12454,12456,12462,12464,12470,12472],{},[604,12453,7297],{},[274,12455,276],{},[52,12457,12458,12460],{},[59,12459,270],{},[59,12461,402],{},[274,12463,7342],{},[52,12465,12466,12468],{},[55,12467,57],{},[59,12469,402],{},[274,12471,276],{},[52,12473,12474,12476],{},[55,12475,57],{},[59,12477,61],{},[63,12479,12480],{"encoding":65},"\\text{Media} = \\frac{1}{2} \\cdot \\frac{\\pi}{2} = \\frac{\\pi}{4}",[33,12482,12484,12505,12588,12671],{"className":12483,"ariaHidden":71},[70],[33,12485,12487,12490,12496,12499,12502],{"className":12486},[75],[33,12488],{"className":12489,"style":7024},[79],[33,12491,12493],{"className":12492},[84,623],[33,12494,7297],{"className":12495},[84],[33,12497],{"className":12498,"style":365},[364],[33,12500,276],{"className":12501},[369],[33,12503],{"className":12504,"style":365},[364],[33,12506,12508,12511,12579,12582,12585],{"className":12507},[75],[33,12509],{"className":12510,"style":291},[79],[33,12512,12514,12517,12576],{"className":12513},[84],[33,12515],{"className":12516},[88,89],[33,12518,12520],{"className":12519},[52],[33,12521,12523,12568],{"className":12522},[96,97],[33,12524,12526,12565],{"className":12525},[101],[33,12527,12529,12543,12551],{"className":12528,"style":310},[105],[33,12530,12531,12534],{"style":109},[33,12532],{"className":12533,"style":114},[113],[33,12535,12537],{"className":12536},[118,119,120,121],[33,12538,12540],{"className":12539},[84,121],[33,12541,402],{"className":12542},[84,121],[33,12544,12545,12548],{"style":130},[33,12546],{"className":12547,"style":114},[113],[33,12549],{"className":12550,"style":138},[137],[33,12552,12553,12556],{"style":141},[33,12554],{"className":12555,"style":114},[113],[33,12557,12559],{"className":12558},[118,119,120,121],[33,12560,12562],{"className":12561},[84,121],[33,12563,270],{"className":12564},[84,121],[33,12566,160],{"className":12567},[159],[33,12569,12571],{"className":12570},[101],[33,12572,12574],{"className":12573,"style":167},[105],[33,12575],{},[33,12577],{"className":12578},[173,89],[33,12580],{"className":12581,"style":2221},[364],[33,12583,7342],{"className":12584},[2225],[33,12586],{"className":12587,"style":2221},[364],[33,12589,12591,12594,12662,12665,12668],{"className":12590},[75],[33,12592],{"className":12593,"style":80},[79],[33,12595,12597,12600,12659],{"className":12596},[84],[33,12598],{"className":12599},[88,89],[33,12601,12603],{"className":12602},[52],[33,12604,12606,12651],{"className":12605},[96,97],[33,12607,12609,12648],{"className":12608},[101],[33,12610,12612,12626,12634],{"className":12611,"style":106},[105],[33,12613,12614,12617],{"style":109},[33,12615],{"className":12616,"style":114},[113],[33,12618,12620],{"className":12619},[118,119,120,121],[33,12621,12623],{"className":12622},[84,121],[33,12624,402],{"className":12625},[84,121],[33,12627,12628,12631],{"style":130},[33,12629],{"className":12630,"style":114},[113],[33,12632],{"className":12633,"style":138},[137],[33,12635,12636,12639],{"style":141},[33,12637],{"className":12638,"style":114},[113],[33,12640,12642],{"className":12641},[118,119,120,121],[33,12643,12645],{"className":12644},[84,121],[33,12646,57],{"className":12647,"style":155},[84,154,121],[33,12649,160],{"className":12650},[159],[33,12652,12654],{"className":12653},[101],[33,12655,12657],{"className":12656,"style":167},[105],[33,12658],{},[33,12660],{"className":12661},[173,89],[33,12663],{"className":12664,"style":365},[364],[33,12666,276],{"className":12667},[369],[33,12669],{"className":12670,"style":365},[364],[33,12672,12674,12677],{"className":12673},[75],[33,12675],{"className":12676,"style":80},[79],[33,12678,12680,12683,12742],{"className":12679},[84],[33,12681],{"className":12682},[88,89],[33,12684,12686],{"className":12685},[52],[33,12687,12689,12734],{"className":12688},[96,97],[33,12690,12692,12731],{"className":12691},[101],[33,12693,12695,12709,12717],{"className":12694,"style":106},[105],[33,12696,12697,12700],{"style":109},[33,12698],{"className":12699,"style":114},[113],[33,12701,12703],{"className":12702},[118,119,120,121],[33,12704,12706],{"className":12705},[84,121],[33,12707,61],{"className":12708},[84,121],[33,12710,12711,12714],{"style":130},[33,12712],{"className":12713,"style":114},[113],[33,12715],{"className":12716,"style":138},[137],[33,12718,12719,12722],{"style":141},[33,12720],{"className":12721,"style":114},[113],[33,12723,12725],{"className":12724},[118,119,120,121],[33,12726,12728],{"className":12727},[84,121],[33,12729,57],{"className":12730,"style":155},[84,154,121],[33,12732,160],{"className":12733},[159],[33,12735,12737],{"className":12736},[101],[33,12738,12740],{"className":12739,"style":167},[105],[33,12741],{},[33,12743],{"className":12744},[173,89],[15,12746,12747,12748,12756,12757,12785,12786,12789],{},"Esta conexión fue ",[19,12749,12752,12753],{"href":12750,"rel":12751},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2602.14487",[23],"descubierta en 2025 por el matemático ",[237,12754,12755],{},"Jim Propp",". Y lo que la hace extraordinaria no es que ",[33,12758,12760,12773],{"className":12759},[36],[33,12761,12763],{"className":12762},[40],[42,12764,12765],{"xmlns":44},[46,12766,12767,12771],{},[49,12768,12769],{},[55,12770,57],{},[63,12772,194],{"encoding":65},[33,12774,12776],{"className":12775,"ariaHidden":71},[70],[33,12777,12779,12782],{"className":12778},[75],[33,12780],{"className":12781,"style":204},[79],[33,12783,57],{"className":12784,"style":155},[84,154]," aparezca en una serie infinita — eso es habitual. Lo extraordinario es que una serie nacida de un proceso tan elemental como lanzar monedas resulte ser ",[210,12787,12788],{},"exactamente"," la misma serie que define una función trigonométrica.",[15,12791,12792],{},"No es una aproximación ni una coincidencia numérica — es una identidad matemática exacta.",[15,12794,12795,12796,12824,12825,11813],{},"Eso sí, no intentes calcular ",[33,12797,12799,12812],{"className":12798},[36],[33,12800,12802],{"className":12801},[40],[42,12803,12804],{"xmlns":44},[46,12805,12806,12810],{},[49,12807,12808],{},[55,12809,57],{},[63,12811,194],{"encoding":65},[33,12813,12815],{"className":12814,"ariaHidden":71},[70],[33,12816,12818,12821],{"className":12817},[75],[33,12819],{"className":12820,"style":204},[79],[33,12822,57],{"className":12823,"style":155},[84,154]," de esta forma. Como el propio Propp ",[19,12826,12828],{"href":12750,"rel":12827},[23],"explicó en el propio documento de Arxiv",[12830,12831,12832],"blockquote",{},[15,12833,12834],{},"Para obtener pi con la precisión de 3,14, podrían hacer falta hasta un billón de lanzamientos de moneda. En parte porque las secuencias pueden alargarse mucho antes de que las caras superen a las cruces — tanto que el valor esperado de la longitud de una secuencia es ¡infinito! Además, no se pueden lanzar todas las monedas a la vez como se pueden soltar agujas: el orden de caras y cruces importa.",[559,12836,12838],{"id":12837},"reflexión-final","Reflexión final",[15,12840,12841],{},"En el artículo anterior, descubrimos que el triángulo de Tartaglia conectaba la combinatoria, el álgebra y la probabilidad. Tres lenguajes que contaban la misma historia.",[15,12843,12844,12845,174],{},"Hoy hemos añadido un cuarto personaje a la trama: el número ",[33,12846,12848,12861],{"className":12847},[36],[33,12849,12851],{"className":12850},[40],[42,12852,12853],{"xmlns":44},[46,12854,12855,12859],{},[49,12856,12857],{},[55,12858,57],{},[63,12860,194],{"encoding":65},[33,12862,12864],{"className":12863,"ariaHidden":71},[70],[33,12865,12867,12870],{"className":12866},[75],[33,12868],{"className":12869,"style":204},[79],[33,12871,57],{"className":12872,"style":155},[84,154],[15,12874,12875,12876,12904],{},"Lanzar una moneda parece un proceso puramente aleatorio. El triángulo de Tartaglia parece pura aritmética. Los números de Catalan parecen un capricho de la combinatoria. Y ",[33,12877,12879,12892],{"className":12878},[36],[33,12880,12882],{"className":12881},[40],[42,12883,12884],{"xmlns":44},[46,12885,12886,12890],{},[49,12887,12888],{},[55,12889,57],{},[63,12891,194],{"encoding":65},[33,12893,12895],{"className":12894,"ariaHidden":71},[70],[33,12896,12898,12901],{"className":12897},[75],[33,12899],{"className":12900,"style":204},[79],[33,12902,57],{"className":12903,"style":155},[84,154]," parece pertenecer exclusivamente a la geometría — a los círculos, a las curvas, al mundo continuo.",[15,12906,12907],{},"Y sin embargo, todos terminan conectados en una sola ecuación.",[15,12909,12910,12911,12939],{},"Hay algo profundamente satisfactorio en descubrir que un hecho nuevo sobre ",[33,12912,12914,12927],{"className":12913},[36],[33,12915,12917],{"className":12916},[40],[42,12918,12919],{"xmlns":44},[46,12920,12921,12925],{},[49,12922,12923],{},[55,12924,57],{},[63,12926,194],{"encoding":65},[33,12928,12930],{"className":12929,"ariaHidden":71},[70],[33,12931,12933,12936],{"className":12932},[75],[33,12934],{"className":12935,"style":204},[79],[33,12937,57],{"className":12938,"style":155},[84,154]," se escondía, desde siempre, detrás de algo tan mundano como lanzar una moneda. Y que nadie lo notó hasta 2025.",[15,12941,12942],{},"Las matemáticas siguen teniendo secretos. Y a veces, están más cerca de lo que pensamos.",[15,12944,12945,12946,12950],{},"Pon la teoría en práctica: ",[19,12947,12949],{"href":576,"rel":12948},[23],"lanza una moneda online"," y observa los patrones en acción.",{"title":1289,"searchDepth":12952,"depth":12952,"links":12953},2,[12954,12962,12963,12964,12965,12966],{"id":561,"depth":12952,"text":562,"children":12955},[12956,12958,12959,12960,12961],{"id":582,"depth":12957,"text":583},3,{"id":850,"depth":12957,"text":851},{"id":1264,"depth":12957,"text":1265},{"id":2152,"depth":12957,"text":2153},{"id":2163,"depth":12957,"text":2164},{"id":4072,"depth":12952,"text":4073},{"id":6151,"depth":12952,"text":6152},{"id":6357,"depth":12952,"text":6358},{"id":9334,"depth":12952,"text":9335},{"id":12837,"depth":12952,"text":12838},"2026-05-27T10:00:00Z","Descubre cómo lanzar una moneda hasta que haya más caras que cruces conduce, inesperadamente, al número π — a través de los números de Catalan y la serie del arcoseno.","md",{"excerpt":12971},{"type":8,"value":12972},[12973,12975,13080,13112],[11,12974,5],{"id":13},[15,12976,17,12977,25,12980,31,12983,174],{},[19,12978,24],{"href":21,"rel":12979},[23],[19,12981,30],{"href":28,"rel":12982},[23],[33,12984,12986,13003],{"className":12985},[36],[33,12987,12989],{"className":12988},[40],[42,12990,12991],{"xmlns":44},[46,12992,12993,13001],{},[49,12994,12995],{},[52,12996,12997,12999],{},[55,12998,57],{},[59,13000,61],{},[63,13002,66],{"encoding":65},[33,13004,13006],{"className":13005,"ariaHidden":71},[70],[33,13007,13009,13012],{"className":13008},[75],[33,13010],{"className":13011,"style":80},[79],[33,13013,13015,13018,13077],{"className":13014},[84],[33,13016],{"className":13017},[88,89],[33,13019,13021],{"className":13020},[52],[33,13022,13024,13069],{"className":13023},[96,97],[33,13025,13027,13066],{"className":13026},[101],[33,13028,13030,13044,13052],{"className":13029,"style":106},[105],[33,13031,13032,13035],{"style":109},[33,13033],{"className":13034,"style":114},[113],[33,13036,13038],{"className":13037},[118,119,120,121],[33,13039,13041],{"className":13040},[84,121],[33,13042,61],{"className":13043},[84,121],[33,13045,13046,13049],{"style":130},[33,13047],{"className":13048,"style":114},[113],[33,13050],{"className":13051,"style":138},[137],[33,13053,13054,13057],{"style":141},[33,13055],{"className":13056,"style":114},[113],[33,13058,13060],{"className":13059},[118,119,120,121],[33,13061,13063],{"className":13062},[84,121],[33,13064,57],{"className":13065,"style":155},[84,154,121],[33,13067,160],{"className":13068},[159],[33,13070,13072],{"className":13071},[101],[33,13073,13075],{"className":13074,"style":167},[105],[33,13076],{},[33,13078],{"className":13079},[173,89],[15,13081,177,13082,208,13110,213],{},[33,13083,13085,13098],{"className":13084},[36],[33,13086,13088],{"className":13087},[40],[42,13089,13090],{"xmlns":44},[46,13091,13092,13096],{},[49,13093,13094],{},[55,13095,57],{},[63,13097,194],{"encoding":65},[33,13099,13101],{"className":13100,"ariaHidden":71},[70],[33,13102,13104,13107],{"className":13103},[75],[33,13105],{"className":13106,"style":204},[79],[33,13108,57],{"className":13109,"style":155},[84,154],[210,13111,212],{},[15,13113,216,13114,222],{},[19,13115,221],{"href":219,"rel":13116},[23],true,"\u002Fblog\u002Fpi-coin-flips-catalan-numbers.es",{"title":5,"description":12968},"lanzando-una-moneda-hasta-obtener-pi","blog\u002Fpi-coin-flips-catalan-numbers.es","tftvlIus-Vzedr65fsds1GlLWOwi3lEU2RboB8VhTN4",{"en":13124,"es":13120},"flipping-a-coin-until-you-get-pi",{"en":13124,"es":13120},1779926067869]