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%I #25 Feb 12 2023 20:51:46
%S 3,17,41,373,823,3565,7625,129293,272171,1139735,2376047,19743201,
%T 40890483,168947957,348259369,11464229693,23547218611,96587303059,
%U 197831583443,1618881562939,3308327420393,13508555185547,27554570432479,449278087454089
%N Numerator 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) = numerator(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 A322755(n): return (n-Fraction(1,2)+Fraction(1<<(m:=n<<1)-1,comb(m,n))).numerator # _Chai Wah Wu_, Feb 12 2023
%Y Cf. A322756.
%K nonn,frac
%O 1,1
%A _N. J. A. Sloane_, Dec 27 2018