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 = 6 and r = 2.
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
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)).
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(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)
-- Reinhard Zumkeller, Sep 19 2014
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Charles Jwo-Yue Lien, Sep 15 2014
STATUS
approved