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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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified August 19 13:44 EDT 2022. Contains 356228 sequences. (Running on oeis4.)