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A247064
Dynamic Betting Game D(n,5,4).
8
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 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, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64
OFFSET
1,2
COMMENTS
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 = 4.
LINKS
Charles Jwo-Yue Lien, Dynamic Betting Game, Southeast Asian Bulletin of Mathematics, 2015, Vol. 39 Issue 6, pp. 799-814.
FORMULA
a(n) = floor(n*32/31).
G.f.: t^31/((1-t)*(1-t^31)) + t/(1-t)^2. - Robert Israel, Sep 11 2014
EXAMPLE
In the case of n=31: For the 1st round, player A bets 1. If A loses, A will end up with D(30,4,3)=32 per reference A247160. If A wins, he has 32 and will bet 0 for the remaining rounds. So he will end up 32. If A does not follow the proposed bet, he will have fewer than 32 at the end. So a(31) = 32.
PROG
(PARI)
vector(100, n, floor(32*n/31)) \\ Derek Orr, Sep 11 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved