A391792 a(n)/3^(n-1) is the expected win if one of three baskets is chosen randomly and the player optimally chooses the coins with values from 1 to n (see Comments for details).

1, 7, 40, 192, 837, 3423, 13478, 51396, 190809, 693595, 2477822, 8740848, 30419671, 104752091, 357748926, 1211285880, 4074000387, 13615817865

Offset

1,2

Comments

At each of n turns one of three 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)/3^(n-1) is the maximum expectation value a player can achieve with the optimal strategy.

Links

Table of n, a(n) for n=1..18.

Ruediger Jehn, Julia program for any number of baskets

Example

a(3)=40. The player shall put coin 2 in the first basket. If the same basket is selected in turn 2 (probability is 1/3), the player shall add coin 3 in this basket, giving an average of 16/3 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 points after the last turn. The expected win is 16/3 * 1/3 + 4 * 2/3 which is 40/3^2.

Crossrefs

Cf. A391537, A391538, A391791 (same game with 2 baskets).

Sequence in context: A263473 A055282 A252816 * A093737 A268402 A351529

Adjacent sequences: A391789 A391790 A391791 * A391793 A391794 A391795

Keyword

nonn,hard,more

Author

Ruediger Jehn, Dec 31 2025

Status

approved