%I #35 Apr 25 2023 17:02:52
%S 1,2,3,4,5,6,8,9,10,11,12,13,16,17,18,19,20,21,22,24,25,26,27,28,29,
%T 32,33,34,35,36,37,38,40,41,42,43,44,45,48,49,50,51,52,53,54,56,57,58,
%U 59,60,61,64,65,66,67,68,69,70,72,73,74,75,76,77,80,81,82,83,84,85,86,88,89,90,91,92,93,96
%N Dynamic Betting Game D(n,5,3).
%C 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 = 5 and r = 3.
%H Charles JwoYue Lien, <a href="http://www.seamsbullmath.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. 799814.
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,0,0,1,1).
%F With a(0)=0, a(n+1)a(n) is a periodic function of n with value = 1,1,1,1,1,1,2,1,1,1,1,1,3.
%F a(n) = a(n1) + a(n13)  a(n14).  _Colin Barker_, Sep 11 2014
%F G.f.: x*(3*x^12 +x^11 +x^10 +x^9 +x^8 +x^7 +2*x^6 +x^5 +x^4 +x^3 +x^2 +x +1) / ((x 1)^2*(x^12 +x^11 +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
%e In the case of n=7: For the 1st round, player A bets 1. If A loses, A will end up with D(6,4,2)=8 per reference A247161. If A wins, he has 8 and will end up with D(8,4,3)=8 per reference A247160. If A does not follow the proposed bet, he will have fewer than 8 at the end. So a(7) = 8.
%Y Cf. A247060, A247061, A247062, A247064, A247160, A247161.
%K nonn,easy
%O 1,2
%A _Charles JwoYue Lien_, Sep 10 2014
