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%I #40 Apr 25 2023 17:22:24
%S 1,8,16,17,24,32,33,40,48,49,56,64,65,72,80,81,88,96,97,104,112,113,
%T 120,128,129,136,144,145,152,160,161,168,176,177,184,192,193,200,208,
%U 209,216,224,225,232,240,241,248,256
%N Dynamic Betting Game D(n,5,1).
%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 = 5 and r = 1.
%H Reinhard Zumkeller, <a href="/A247061/b247061.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_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F With a(0)=0, a(n+1)-a(n) is a periodic function of n with value = 1, 7, 8.
%F a(n) = a(n-1) + a(n-3) - a(n-4). - _Colin Barker_, Sep 11 2014
%F G.f.: x*(8*x^2+7*x+1) / ((x-1)^2*(x^2+x+1)). - _Colin Barker_, Sep 11 2014
%e In the case of n=3: For the 1st round, player A bets 2. If A loses, A will end up with 16 by betting all he has for the last 4 rounds. If A wins, he has 5 and will end up with D(5,4,1)=16 per reference A247060. If A does not follow the proposed bet, he will have fewer than 16 at the end. So a(3) = 16.
%o (PARI) Vec(x*(8*x^2+7*x+1)/((x-1)^2*(x^2+x+1)) + O(x^100)) \\ _Colin Barker_, Sep 11 2014
%o (Haskell)
%o a247061 n = a247061_list !! (n-1)
%o a247061_list = [1,8,16,17] ++ zipWith (+)
%o (drop 3 a247061_list) (zipWith (-) (tail a247061_list) a247061_list)
%o -- _Reinhard Zumkeller_, Sep 19 2014
%Y Cf. A247060, A247062, A247063, A247064, A247160, A247161.
%K nonn,easy
%O 1,2
%A _Charles Jwo-Yue Lien_, Sep 10 2014
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