A366995 a(n) is the numerator of the expected end time of a game with three gamblers, one of which starts with capital n, the others with capital 1 each, conditional on the event that one of the two poor players wins.

3, 39, 359, 2477, 119667, 1522705, 46419629, 6143100517, 5472109127035, 790136773603303, 278129286200597661, 16684426086791338103, 503067648850136040148699, 2626565018569118643191009, 10920130209346850287269887104735, 236686188450953790757840351941895

Offset

1,1

Comments

In each round of the game, 1 unit is transferred from one randomly chosen player to another. Players play until they are out of money, so when the first player is out the other two continue to play. The winner is the player who ends up with all n+2 units of money.

Links

Pontus von Brömssen, Table of n, a(n) for n = 1..53

Persi Diaconis and Stewart N. Ethier, Gambler’s ruin and the ICM, Statist. Sci. 37 (3) 289-305, August 2022.

Persi Diaconis, Gambler's ruin with k gamblers, slides from talk in the Rutgers Experimental Mathematics Seminar, Fall 2023 Semester, Oct 12, 2023.

Experimental Mathematics, Gambler’s ruin with k gamblers, recording of talk, Vimeo video, Oct 22, 2023.

Prog

(Sage)
from itertools import permutations
def T(n):
    nodes = [(i, j) for i in range(n+2) for j in range((n+2-i)//2+1)]
    m = len(nodes)
    Q0 = {x:{y:0 for y in nodes} for x in nodes}
    for x in nodes:
        c1 = x+(n+2-sum(x), )
        for i, j in permutations(range(3), int(2)):
            if c1[i] and c1[j]:
                c2 = list(c1)
                c2[i] -= 1
                c2[j] += 1
                y = (c2[0], min(c2[1:]))
                if c2[0] != n+2:
                    Q0[x][y] += n+2-c2[0]
    Q0 = matrix(QQ, [list(R.values()) for R in Q0.values()])
    s = sum(Q0.columns())
    Q = identity_matrix(QQ, m-1)
    for i in range(1, m):
        for j in range(1, m):
            if s[i] != 0: Q[i-1, j-1] -= Q0[i, j]/s[i]
    return (Q**(-1)*ones_matrix(QQ, m-1))[-2, 0]
def A366995(n):
    return T(n).numerator()
def A366996(n):
    return T(n).denominator()

Crossrefs

Cf. A366566 (a(n)/A366996(n) rounded to nearest integer), A366996 (denominators).

Sequence in context: A342969 A050392 A292294 * A191468 A203243 A063035

Adjacent sequences: A366992 A366993 A366994 * A366996 A366997 A366998

Keyword

nonn,frac

Author

Pontus von Brömssen, Oct 31 2023

Status

approved