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A247065
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Dynamic Betting Game D(n,6,1).
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2
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1, 16, 24, 32, 40, 49, 64, 65, 80, 88, 96, 104, 113, 128, 129, 144, 152, 160, 168, 177, 192, 193, 208, 216, 224, 232, 241, 256, 257, 272, 280, 288, 296, 305, 320, 321, 336, 344, 352, 360, 369, 384, 385, 400, 408, 416, 424, 433, 448, 449, 464, 472, 480, 488, 497, 512, 513, 528
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OFFSET
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1,2
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COMMENTS
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Players A and B bet in a k-round game. Player A has an initial amount of money n. In each round, player A can wager an integer between 0 and what he has. Player A then gains or loses an amount equal to his wager depending on whether player B lets him win or lose. Player B tries to minimize player A's money at the end. The number of rounds player A can lose is r. a(n) is the maximum amount of money player A can have at the end of the game for k = 6 and r = 1.
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LINKS
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Charles Jwo-Yue Lien, Dynamic Betting Game, Southeast Asian Bulletin of Mathematics, 2015, Vol. 39 Issue 6, pp. 799-814.
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FORMULA
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With a(0)=0, a(n+1)-a(n) is a periodic function of n with value = 1,15,8,8,8,9,15.
G.f.:x*(1+x)*(1+14*x-6*x^2+14*x^3-6*x^4+15*x^5)/((1-x)^2*(1+x+x^2+x^3+x^4+x^5+x^6)).
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EXAMPLE
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In the case of n=3: For the first round, player A bets 2. Player B will let player A win. Otherwise player A will end up with 32 by betting all he has for the last 5 rounds. Therefore after the first round, player A has 5 and will end up with D(5,5,1)=24 per reference A247061. If A does not follow the proposed bet, he will have fewer than 24 at the end. So a(3) = 24.
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PROG
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(PARI) Vec(x*(x+1)*(15*x^5-6*x^4+14*x^3-6*x^2+14*x+1)/((x-1)^2*(x^6+x^5+x^4+x^3+x^2+x+1)) + O(x^100)) \\ Colin Barker, Sep 14 2014
(Haskell)
a247065 n = a247065_list !! (n-1)
a247065_list = [1, 16, 24, 32, 40, 49, 64, 65] ++ zipWith (+)
(drop 7 a247065_list) (zipWith (-) (tail a247065_list) a247065_list)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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