%I #27 Nov 01 2023 10:01:37
%S 3,6,9,13,17,22,28,34,41,49,58,67,76,87,98,109,122,135,149,163,178,
%T 194,210,227,245,263,282,302,322,343,365,387,410,434,458,483,509,535,
%U 562,590,619,648,677,708,739,770,803,836,869,904,939,974,1011,1048,1085
%N 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 For details see the Diaconis link.
%C Initially, terms up to a(25) were calculated using Monte Carlo simulation of 10^9 games at each value of n.
%C 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 The expected shorter end time also allowing the rich player to win would be 2*n+1 (Bachelier, 1912, page 149).
%H Louis Bachelier, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k9804939z">Calcul des probabilités. Tome I</a>, Gauthier-Villars, Paris, 1912.
%H 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 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 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 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 Hugo Pfoertner, <a href="/A366566/a366566_2.png">Distribution of the number of games won, n=3</a>, plotted vs end time.
%H Hugo Pfoertner, <a href="/A366566/a366566_3.png">Distribution of the number of games won, n=5</a>, plotted vs end time.
%H Hugo Pfoertner, <a href="/A366566/a366566_4.png">Distribution of the number of games won, n=6</a>, plotted vs end time.
%H Hugo Pfoertner, <a href="/A366566/a366566_5.png">Distribution of the number of games won, n=10</a>, plotted vs end time.
%H Hugo Pfoertner, <a href="/A366566/a366566_6.png">Distribution of the number of games won, n=20</a>, plotted vs end time.
%F a(n) equals A366995(n)/A366996(n) rounded to the nearest integer. - _Pontus von Brömssen_, Oct 31 2023
%Y Cf. A366166, A366567 (mode of corresponding probability distributions), A366995, A366996.
%K nonn
%O 1,1
%A _Hugo Pfoertner_, Oct 13 2023
%E a(26)-a(55) from _Pontus von Brömssen_, Oct 31 2023