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A247063
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Dynamic Betting Game D(n,5,3).
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8
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1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 93, 96
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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 = 5 and r = 3.
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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.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,1,-1).
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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 = 1,1,1,1,1,1,2,1,1,1,1,1,3.
a(n) = a(n-1) + a(n-13) - a(n-14). - Colin Barker, Sep 11 2014
G.f.: x*(3*x^12 +x^11 +x^10 +x^9 +x^8 +x^7 +2*x^6 +x^5 +x^4 +x^3 +x^2 +x +1) / ((x -1)^2*(x^12 +x^11 +x^10 +x^9 +x^8 +x^7 +x^6 +x^5 +x^4 +x^3 +x^2 +x +1)). - Colin Barker, Sep 11 2014
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EXAMPLE
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In the case of n=7: For the 1st round, player A bets 1. If A loses, A will end up with D(6,4,2)=8 per reference A247161. If A wins, he has 8 and will end up with D(8,4,3)=8 per reference A247160. If A does not follow the proposed bet, he will have fewer than 8 at the end. So a(7) = 8.
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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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STATUS
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approved
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