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.
KEYWORD
nonn
AUTHOR
Wing Hong Tony Wong, Jiao Xu, Dec 18 2015
STATUS
approved