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%I #30 Apr 25 2023 19:14:14
%S 1,2,6,8,12,16,17,21,24,27,32,33,34,38,40,44,48,49,53,56,59,64,65,66,
%T 70,72,76,80,81,85,88,91,96,97,98,102,104,108,112,113,117,120,123,128,
%U 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
%N Dynamic Betting Game D(n,6,2).
%C 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.
%H Reinhard Zumkeller, <a href="/A247066/b247066.txt">Table of n, a(n) for n = 1..1000</a>
%H Charles Jwo-Yue Lien, <a href="http://www.seams-bull-math.ynu.edu.cn/quick_search_result.jsp?search&cond=Dynamic%20Betting%20Game">Dynamic Betting Game</a>, Southeast Asian Bulletin of Mathematics, 2015, Vol. 39 Issue 6, pp. 799-814.
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,1,-1).
%F 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.
%F a(n) = a(n-1)+a(n-11)-a(n-12).
%F 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)).
%e 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.
%t 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 *)
%o (Haskell)
%o a247066 n = a247066_list !! (n-1)
%o a247066_list = [1,2,6,8,12,16,17,21,24,27,32,33] ++ zipWith (+)
%o (drop 11 a247066_list) (zipWith (-) (tail a247066_list) a247066_list)
%o -- _Reinhard Zumkeller_, Sep 19 2014
%o (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
%Y Cf. A247061, A247062, A247063, A247064, A247065, A247160, A247161.
%K nonn
%O 1,2
%A _Charles Jwo-Yue Lien_, Sep 15 2014
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