A247066 Dynamic Betting Game D(n,6,2).

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

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

Cf. A247061, A247062, A247063, A247064, A247065, A247160, A247161.

Sequence in context: A268177 A083769 A057656 * A084724 A283233 A189400

Adjacent sequences: A247063 A247064 A247065 * A247067 A247068 A247069

Keyword

nonn

Author

Charles Jwo-Yue Lien, Sep 15 2014

Status

approved