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A247062
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Dynamic Betting Game D(n,5,2).
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9
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1, 2, 5, 6, 8, 11, 12, 16, 17, 18, 21, 22, 24, 27, 28, 32, 33, 34, 37, 38, 40, 43, 44, 48, 49, 50, 53, 54, 56, 59, 60, 64, 65, 66, 69, 70, 72, 75, 76, 80, 81, 82, 85, 86, 88, 91, 92, 96, 97, 98, 101, 102, 104, 107, 108, 112
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refs;
listen;
history;
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internal format)
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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 = 2.
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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 = 1,1,3,1,2,3,1,4.
a(n) = a(n-1) + a(n-8) - a(n-9). - Colin Barker, Sep 11 2014
G.f.: x*(4*x^7+x^6+3*x^5+2*x^4+x^3+3*x^2+x+1) / ((x-1)^2*(x+1)*(x^2+1)*(x^4+1)). - Colin Barker, Sep 11 2014
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EXAMPLE
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In the case of n=3: For the 1st round, player A bets 1. If A loses, A will end up with D(2,4,1)=5 per reference A247060. If A wins, he will end up with D(4,4,2)=5 per reference A247161. If A does not follow the proposed bet, he will have fewer than 5 at the end. So a(3) = 5.
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MATHEMATICA
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LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 1, -1}, {1, 2, 5, 6, 8, 11, 12, 16, 17}, 60] (* Harvey P. Dale, Nov 21 2020 *)
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PROG
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(PARI) Vec(x*(4*x^7+x^6+3*x^5+2*x^4+x^3+3*x^2+x+1)/((x-1)^2*(x+1)*(x^2+1)*(x^4+1)) + O(x^100)) \\ Colin Barker, Sep 11 2014
(Haskell)
a247062 n = a247062_list !! (n-1)
a247062_list = [1, 2, 5, 6, 8, 11, 12, 16, 17] ++ zipWith (+)
(drop 8 a247062_list) (zipWith (-) (tail a247062_list) a247062_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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STATUS
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approved
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