

A322755


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


1



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

1,1


COMMENTS

A 2ncard playing deck is shuffled and then revealed onebyone 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. 234235. Solution by John Henry Steelman.


FORMULA

The optimal payoff is n  1/2 + 2^(2n1)/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*n1)/binomial(2*n, n)); \\ Michel Marcus, Dec 28 2018


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



