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A322756
Denominator of expected payoff in the "Guessing Card Colors" game with a 2n-card deck, using an optimal strategy.
2
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
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
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane, Dec 27 2018
STATUS
approved