This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A366566 #27 Nov 01 2023 10:01:37 %S A366566 3,6,9,13,17,22,28,34,41,49,58,67,76,87,98,109,122,135,149,163,178, %T A366566 194,210,227,245,263,282,302,322,343,365,387,410,434,458,483,509,535, %U A366566 562,590,619,648,677,708,739,770,803,836,869,904,939,974,1011,1048,1085 %N A366566 a(n) is the expected end time of a game with three gamblers, one of which starts with capital n, the others with capital 1 each. The end time, rounded to the nearest integer, is given for games in which one of the two poor players wins. %C A366566 For details see the Diaconis link. %C A366566 Initially, terms up to a(25) were calculated using Monte Carlo simulation of 10^9 games at each value of n. %C A366566 The expected end times without rounding to nearest integer are: 3.00, 5.57, 8.76, 12.57, 17.03, 22.14, 27.91, 34.33, 41.41, 49.15, 57.55, 66.61, 76.33, 86.72, ... . %C A366566 The expected shorter end time also allowing the rich player to win would be 2*n+1 (Bachelier, 1912, page 149). %H A366566 Louis Bachelier, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k9804939z">Calcul des probabilités. Tome I</a>, Gauthier-Villars, Paris, 1912. %H A366566 Persi Diaconis and Stewart N. Ethier, <a href="https://doi.org/10.1214/21-STS826">Gambler’s Ruin and the ICM</a>, Statist. Sci. 37 (3) 289 - 305, August 2022. %H A366566 Persi Diaconis, <a href="https://sites.math.rutgers.edu/~zeilberg/expmath/diaconis23.pdf">Gambler's ruin with k gamblers</a> (slide 3), talk in the Rutgers Experimental Mathematics Seminar, <a href="https://sites.math.rutgers.edu/~zeilberg/expmath/archive23.html">Fall 2023 Semester</a>, Oct. 12, 2023. %H A366566 Experimental Mathematics, <a href="https://vimeo.com/876882233">GAMBLER’S RUIN WITH K GAMBLERS</a>, recording of talk, Vimeo video (time after 11:55), Oct 22, 2023. %H A366566 Hugo Pfoertner, <a href="/A366566/a366566_1.png">Example of the time history of a game with n=3</a>, i.e., the "rich" player starts with 3 chips. %H A366566 Hugo Pfoertner, <a href="/A366566/a366566_2.png">Distribution of the number of games won, n=3</a>, plotted vs end time. %H A366566 Hugo Pfoertner, <a href="/A366566/a366566_3.png">Distribution of the number of games won, n=5</a>, plotted vs end time. %H A366566 Hugo Pfoertner, <a href="/A366566/a366566_4.png">Distribution of the number of games won, n=6</a>, plotted vs end time. %H A366566 Hugo Pfoertner, <a href="/A366566/a366566_5.png">Distribution of the number of games won, n=10</a>, plotted vs end time. %H A366566 Hugo Pfoertner, <a href="/A366566/a366566_6.png">Distribution of the number of games won, n=20</a>, plotted vs end time. %F A366566 a(n) equals A366995(n)/A366996(n) rounded to the nearest integer. - _Pontus von Brömssen_, Oct 31 2023 %Y A366566 Cf. A366166, A366567 (mode of corresponding probability distributions), A366995, A366996. %K A366566 nonn %O A366566 1,1 %A A366566 _Hugo Pfoertner_, Oct 13 2023 %E A366566 a(26)-a(55) from _Pontus von Brömssen_, Oct 31 2023