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 = 4 and r = 2.
With a(0) = 0, a(n+1) - a(n) is a periodic function of n with value = 1,1,2,1,1,2,1,2,1,1,3.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
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).
FORMULA
a(n) = floor(12*n/11) + floor((4*n+1)/11).
a(n) = a(n-1) + a(n-11) - a(n-12). - Colin Barker, Sep 11 2014
G.f.: x*(3*x^10 +x^9 +x^8 +2*x^7 +x^6 +2*x^5 +x^4 +x^3 +2*x^2 +x +1) / ((x -1)^2*(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
PROG
(PARI)
vector(100, n, floor(12*n/11)+floor((4*n+1)/11) \\ Derek Orr, Sep 11 2014
(Haskell)
a247161 n = a247161_list !! (n-1)
a247161_list = [1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 16, 17] ++ zipWith (+)
(drop 11 a247161_list) (zipWith (-) (tail a247161_list) a247161_list)
-- Reinhard Zumkeller, Sep 19 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Charles Jwo-Yue Lien, Sep 10 2014
STATUS
approved