A322756 Denominator of expected payoff in the "Guessing Card Colors" game with a 2n-card deck, using an optimal strategy.

2, 6, 10, 70, 126, 462, 858, 12870, 24310, 92378, 176358, 1352078, 2600150, 10029150, 19389690, 601080390, 1166803110, 4537567650, 8836315950, 68923264410, 134564468610, 526024740930, 1029178840950, 16123801841550, 31602651609438, 123979633237026

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..26.

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) = denominator(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 A322756(n): return (n-Fraction(1, 2)+Fraction(1<<(m:=n<<1)-1, comb(m, n))).denominator # Chai Wah Wu, Feb 12 2023

Crossrefs

Cf. A322755.

This is twice A001790.

Sequence in context: A324547 A093880 A080397 * A048782 A358739 A083458

Adjacent sequences: A322753 A322754 A322755 * A322757 A322758 A322759

Keyword

nonn,frac

Author

N. J. A. Sloane, Dec 27 2018

Status

approved