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A265920 Numerator of the probability that Alice wins the following game: Alice and Bob take turn (Alice starts first) to gain 1 or 2 chips randomly and independently with 1/2 chance, and the first player that collects at least n chips is the winner. 1
1, 3, 13, 47, 185, 723, 2821, 11127, 43825, 173147, 685181, 2713919, 10762793, 42715619, 169654133, 674238983, 2680944545, 10665068907, 42443750893, 168973210575, 672913173913, 2680539263219, 10680581419493, 42566341729431, 169678604019217, 676501889994363 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The formula is proved by Wing Hong Tony Wong and Jiao Xu, and then proved by Taoye Zhang and Ju Zhou independently.
The probability that Alice wins the game is a(n)/4^(n-1).
LINKS
FORMULA
a(n) = (4^n + Sum_(k=1..n) 4^(n-k)*(binomial(k, n-k)+binomial(k-1, n-k))^2)/8.
MATHEMATICA
Table[(4^n + Sum[4^(n - k)*(Binomial[k, n - k] + Binomial[k - 1, n - k])^2, {k, n}])/8, {n, 100}]
CROSSREFS
A265919 provides the integer sequence of the numerator of the probability that Bob wins the game. The corresponding terms of these two sequences add up to 4^n.
Sequence in context: A228529 A084519 A304628 * A262322 A180278 A193164
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 28 16:58 EDT 2024. Contains 371254 sequences. (Running on oeis4.)