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A247160
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Dynamic Betting Game D(n,4,3).
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9
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80
(list;
graph;
refs;
listen;
history;
text;
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 = 4 and r = 3. Note that with a(0)=0, a(n+1)-a(n) is a periodic function of n with value = 1,1,1,1,1,1,1,1,1,1,1,1,1,1,2.
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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 (1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).
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FORMULA
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a(n) = floor(n*16/15).
a(n) = a(n-1) + a(n-15) - a(n-16). - Colin Barker, Sep 11 2014
G.f.: x*(2*x^14 +x^13 +x^12 +x^11 +x^10 +x^9 +x^8 +x^7 +x^6 +x^5 +x^4 +x^3 +x^2 +x +1) / ((x -1)^2*(x^2 +x +1)*(x^4 +x^3 +x^2 +x +1)*(x^8 -x^7 +x^5 -x^4 +x^3 -x +1)). - Colin Barker, Sep 11 2014
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PROG
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(Haskell)
a247160 n = a247160_list !! (n-1)
a247160_list = [1..14] ++ [16, 17] ++ zipWith (+)
(drop 15 a247160_list) (zipWith (-) (tail a247160_list) a247160_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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