

A247066


Dynamic Betting Game D(n,6,2).


2



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

1,2


COMMENTS

Players A and B bet in a kround 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

Charles JwoYue Lien, Dynamic Betting Game, Southeast Asian Bulletin of Mathematics, 2015, Vol. 39 Issue 6, pp. 799814.
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(n1)+a(n11)a(n12).
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)/((1x)^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 !! (n1)
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)/((1x)^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



STATUS

approved



