|
%I #21 Feb 13 2023 03:09:29
%S 2,6,10,70,126,462,858,12870,24310,92378,176358,1352078,2600150,
%T 10029150,19389690,601080390,1166803110,4537567650,8836315950,
%U 68923264410,134564468610,526024740930,1029178840950,16123801841550,31602651609438,123979633237026
%N Denominator of expected payoff in the "Guessing Card Colors" game with a 2n-card deck, using an optimal strategy.
%C A 2n-card playing deck is shuffled and then revealed one-by-one to a player who guesses the color (red or black) of each card prior to its being revealed. The player earns one dollar for each card whose color he guesses correctly; there is no penalty for being wrong.
%D Thane Plambeck and others, Posting to Math Fun Mailing List, Dec 26 2018.
%H Michael Andreoli (proposer), <a href="https://www.jstor.org/stable/2687606">Guessing Card Colors, Problem #630</a>, College Mathematics Journal Vol. 30, No. 3 (May, 1999), pp. 234-235. Solution by John Henry Steelman.
%F The optimal payoff is n - 1/2 + 2^(2n-1)/binomial(2n,n).
%e 3/2, 17/6, 41/10, 373/70, 823/126, 3565/462, 7625/858, 129293/12870, 272171/24310, 1139735/92378, 2376047/176358, ...
%o (PARI) a(n) = denominator(n - 1/2 + 2^(2*n-1)/binomial(2*n,n)); \\ _Michel Marcus_, Dec 28 2018
%o (Python)
%o from fractions import Fraction
%o from math import comb
%o def A322756(n): return (n-Fraction(1,2)+Fraction(1<<(m:=n<<1)-1,comb(m,n))).denominator # _Chai Wah Wu_, Feb 12 2023
%Y Cf. A322755.
%Y This is twice A001790.
%K nonn,frac
%O 1,1
%A _N. J. A. Sloane_, Dec 27 2018
|
