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A247066
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Dynamic Betting Game D(n,6,2).
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2
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1, 2, 6, 8, 12, 16, 17, 21, 24, 27, 32, 33, 34, 38, 40, 44, 48, 49, 53, 56, 59, 64, 65, 66, 70, 72, 76, 80, 81, 85, 88, 91, 96, 97, 98, 102, 104, 108, 112, 113, 117, 120, 123, 128, 129, 130, 134, 136, 140, 144, 145, 149, 152, 155, 160, 161, 162, 166, 168, 172, 176, 177, 181, 184, 187, 192, 193, 194, 198
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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 = 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.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,1,-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 = 1,1,4,2,4,4,1,4,3,3,5.
a(n) = a(n-1)+a(n-11)-a(n-12).
G.f.: x*(1+x+4*x^2+2*x^3+4*x^4+4*x^5+x^6+4*x^7+3*x^8+3*x^9+5*x^10)/((1-x)^2*(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10)).
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EXAMPLE
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In the case of n=6: For the first round, player A bets 2. Player B will let player A win. Otherwise player A will end up with D(4,5,1)=17 per reference A247061. Therefore after the first round, player A has 8 and will end up with D(8,5,2)=16 per reference A247062. Alternatively, player A bets 3 for the first round. Player B will let player A lose. Otherwise player A will end up with D(9,5,2)=17 per reference A247062. Therefore after the first round, player A has 3 and will end up with D(3,5,1)=16 per reference A247061. If A does not follow the proposed bets, he will have fewer than 16 at the end. So a(6) = 16.
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MATHEMATICA
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LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {1, 2, 6, 8, 12, 16, 17, 21, 24, 27, 32, 33}, 70] (* Harvey P. Dale, Aug 11 2021 *)
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PROG
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(Haskell)
a247066 n = a247066_list !! (n-1)
a247066_list = [1, 2, 6, 8, 12, 16, 17, 21, 24, 27, 32, 33] ++ zipWith (+)
(drop 11 a247066_list) (zipWith (-) (tail a247066_list) a247066_list)
(PARI) Vec(x*(1+x+4*x^2+2*x^3+4*x^4+4*x^5+x^6+4*x^7+3*x^8+3*x^9+5*x^10)/((1-x)^2*(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10)) + O(x^100)) \\ Altug Alkan, Feb 05 2016
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CROSSREFS
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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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