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.

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

Michael De Vlieger, Table of n, a(n) for n = 1..1662

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

Adjacent sequences: A265916 A265917 A265918 * A265920 A265921 A265922

Keyword

nonn

Author

Wing Hong Tony Wong, Jiao Xu, Dec 18 2015

Status

approved