1,3

Configuration P = (b(1), ..., b(n)) is valid if 0 <= b(i) <= i for 1 <= i <= n and Sum_{i=1..n} b(i) = n.

The identity configuration is I = (1, 1, ..., 1).

A move consists of choosing two bottles i, j and pouring k = min{b(i), j-b(j)} units from i to j.

a(n) is the maximum, over all valid initial configurations P, of the minimum number of moves required to reach I.

Yifan Xie, Proof of the formula

a(n) <= (4*n - 2)/3.

For n = 3 and the initial configuration P = (1, 2, 0), the minimum number of moves required to reach I = (1, 1, 1) is 2:

Move 1: Operate on (1, 3). min{b(1), 3 - b(3)} = 1. b(1) is decreased by 1, becoming 0; b(3) is increased by 1, becoming 1. The configuration becomes (0, 2, 1).

Move 2: Operate on (2, 1). min{b(2), 1 - b(1)} = 1. b(2) is decreased by 1, becoming 1; b(1) is increased by 1, becoming 1. The configuration becomes (1, 1, 1).

Cf. A387661 (ways attained), A000707 (total configurations).

Sequence in context: A051837 A026371 A187335 * A161188 A184117 A184624

Adjacent sequences: A387315 A387316 A387317 * A387319 A387320 A387321

nonn,more

Yifan Xie, Aug 26 2025

a(15) from Sean A. Irvine, Sep 17 2025

approved