A391791 - OEIS

A391791

a(n)/2^(n-1) is the expected win if one of two baskets is chosen randomly and the player optimally chooses the coins with values from 1 to n (see Comments for details).

1, 5, 20, 65, 190, 530, 1407, 3585, 8942, 21819, 52204, 123190, 287044, 661329, 1509949, 3420580, 7688444, 17173896, 38143782, 84262365, 185278522, 405700861

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OFFSET

1,2

COMMENTS

At each of n turns one of two baskets is randomly chosen and the player puts one out of n coins which have the values from 1 to n in that basket. After the last turn, the player may choose which basket they take. a(n)/2^(n-1) is the maximum expectation value a player can achieve with the optimal strategy.

Conjecture: For n>9 the optimal strategy is to put coin floor((n+3)/2) in the first basket.

LINKS

Table of n, a(n) for n=1..22.

Ruediger Jehn, Julia program for any number of baskets

EXAMPLE

a(3)=20. The player shall put coin 2 in the first basket. If the same basket is selected in turn 2, the player shall add coin 3 in this basket, giving an average of 5.5 points after the last turn. If the second basket is selected in turn 2 the player shall put coin 1 in this basket, giving an average of 4.5 points after the last turn. The average of 5.5 and 4.5 is 5 which is 20/2^2.

CROSSREFS

Cf. A391537, A391538, A391792 (same game with 3 baskets).

Sequence in context: A160528 A023004 A001873 * A120297 A271066 A271599

Adjacent sequences: A391788 A391789 A391790 * A391792 A391793 A391794

KEYWORD

nonn,hard,more

AUTHOR

Ruediger Jehn, Dec 20 2025

STATUS

approved