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%I #32 Apr 25 2023 17:21:15
%S 1,16,24,32,40,49,64,65,80,88,96,104,113,128,129,144,152,160,168,177,
%T 192,193,208,216,224,232,241,256,257,272,280,288,296,305,320,321,336,
%U 344,352,360,369,384,385,400,408,416,424,433,448,449,464,472,480,488,497,512,513,528
%N Dynamic Betting Game D(n,6,1).
%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 = 1.
%H Reinhard Zumkeller, <a href="/A247065/b247065.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_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,1,-1).
%F With a(0)=0, a(n+1)-a(n) is a periodic function of n with value = 1,15,8,8,8,9,15.
%F G.f.:x*(1+x)*(1+14*x-6*x^2+14*x^3-6*x^4+15*x^5)/((1-x)^2*(1+x+x^2+x^3+x^4+x^5+x^6)).
%F a(n) = a(n-1)+a(n-7)-a(n-8). - _Colin Barker_, Sep 14 2014
%e In the case of n=3: For the first round, player A bets 2. Player B will let player A win. Otherwise player A will end up with 32 by betting all he has for the last 5 rounds. Therefore after the first round, player A has 5 and will end up with D(5,5,1)=24 per reference A247061. If A does not follow the proposed bet, he will have fewer than 24 at the end. So a(3) = 24.
%o (PARI) Vec(x*(x+1)*(15*x^5-6*x^4+14*x^3-6*x^2+14*x+1)/((x-1)^2*(x^6+x^5+x^4+x^3+x^2+x+1)) + O(x^100)) \\ _Colin Barker_, Sep 14 2014
%o (Haskell)
%o a247065 n = a247065_list !! (n-1)
%o a247065_list = [1,16,24,32,40,49,64,65] ++ zipWith (+)
%o (drop 7 a247065_list) (zipWith (-) (tail a247065_list) a247065_list)
%o -- _Reinhard Zumkeller_, Sep 19 2014
%Y Cf. A247061, A247062, A247063, A247064, A247066, A247160, A247161.
%K nonn,easy
%O 1,2
%A _Charles Jwo-Yue Lien_, Sep 14 2014
%E Typo in data fixed by _Colin Barker_, Sep 14 2014