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A265919
Numerator of the probability that Bob 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.
2
0, 1, 3, 17, 71, 301, 1275, 5257, 21711, 88997, 363395, 1480385, 6014423, 24393245, 98781323, 399502841, 1614022751, 6514800277, 26275725843, 105904696369, 426598453863, 1717507247885, 6911604624923, 27802402448233, 111796372691439, 449398016848261
OFFSET
1,3
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 Bob wins the game is a(n)/4^(n-1).
LINKS
Tony W. H. Wong, Jiao Xu, A Probabilistic Take-Away Game, J. Int. Seq., Vol. 21 (2018), Article 18.6.3.
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
A265920 provides the integer sequence of the numerator of the probability that Alice wins the game. The corresponding terms of these two sequences add up to 4^n.
Sequence in context: A069468 A241776 A270231 * A317452 A128948 A342364
KEYWORD
nonn
AUTHOR
STATUS
approved