A322755 Numerator of expected payoff in the "Guessing Card Colors" game with a 2n-card deck, using an optimal strategy.

3, 17, 41, 373, 823, 3565, 7625, 129293, 272171, 1139735, 2376047, 19743201, 40890483, 168947957, 348259369, 11464229693, 23547218611, 96587303059, 197831583443, 1618881562939, 3308327420393, 13508555185547, 27554570432479, 449278087454089

Offset

1,1

Comments

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.

References

Thane Plambeck and others, Posting to Math Fun Mailing List, Dec 26 2018.

Links

Table of n, a(n) for n=1..24.

Michael Andreoli (proposer), Guessing Card Colors, Problem #630, College Mathematics Journal Vol. 30, No. 3 (May, 1999), pp. 234-235. Solution by John Henry Steelman.

Formula

The optimal payoff is n - 1/2 + 2^(2n-1)/binomial(2n,n).

Example

3/2, 17/6, 41/10, 373/70, 823/126, 3565/462, 7625/858, 129293/12870, 272171/24310, 1139735/92378, 2376047/176358, ...

Prog

(PARI) a(n) = numerator(n - 1/2 + 2^(2*n-1)/binomial(2*n, n)); \\ Michel Marcus, Dec 28 2018
(Python)
from fractions import Fraction
from math import comb
def A322755(n): return (n-Fraction(1, 2)+Fraction(1<<(m:=n<<1)-1, comb(m, n))).numerator # Chai Wah Wu, Feb 12 2023

Crossrefs

Cf. A322756.

Sequence in context: A181981 A089637 A135471 * A226492 A092347 A215429

Adjacent sequences: A322752 A322753 A322754 * A322756 A322757 A322758

Keyword

nonn,frac

Author

N. J. A. Sloane, Dec 27 2018

Status

approved