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A265920
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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.
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1
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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
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OFFSET
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1,2
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COMMENTS
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The probability that Alice wins the game is a(n)/4^(n-1).
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LINKS
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FORMULA
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a(n) = (4^n + Sum_(k=1..n) 4^(n-k)*(binomial(k, n-k)+binomial(k-1, n-k))^2)/8.
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MATHEMATICA
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Table[(4^n + Sum[4^(n - k)*(Binomial[k, n - k] + Binomial[k - 1, n - k])^2, {k, n}])/8, {n, 100}]
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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