A390083 The number of irreducible zero-sum subsets of T(n) = {-2*n+1, -2*n+3, ..., -3, -1, 1, 3, ..., 2*n-3, 2*n-1} that contain -2*n+1 but not 2*n-1.

0, 0, 0, 1, 3, 6, 14, 30, 62, 115, 200, 351, 622, 1125, 2050, 3673, 6351, 10731, 17538, 27758, 42971, 65002, 96469, 141963, 207006, 300417, 436184, 633784, 919745, 1337895, 1941711, 2810990, 4050271, 5803114, 8252575, 11649367, 16300418

Offset

1,5

Comments

A zero-sum subset of T(n) is called irreducible if it doesn't contain any smaller zero-sum subset.

Links

Table of n, a(n) for n=1..37.

Chinese BBS, Balanced samples of the initial segment of natural numbers (in Chinese).

Formula

a(n) = (A389802(2*n) - A389802(2*n-2) - 1) / 2.

a(n) = (A389804(n) - A389804(n-1) - 1) / 2.

Example

For n = 3, a(3) = 0, since {-5, -3, -1, 1, 3, 5} has no such subset.
For n = 4, a(4) = 1, since {-7, -5, -3, -1, 1, 3, 5, 7} has only one such subset: {-1, 3, 5, -7}.
For n = 5, a(5) = 3, since {-9, -7, -5, -3, -1, 1, 3, 5, 7, 9} has 3 such subsets: {1, 3, 5, -9}, {-1, 3, 7, -9}, {-3, 5, 7, -9}.

Crossrefs

Cf. A389802, A389804, A389807, A390082.

Sequence in context: A077067 A083797 A308580 * A394990 A395004 A307457

Adjacent sequences: A390080 A390081 A390082 * A390084 A390085 A390086

Keyword

nonn

Author

Hu Junhua, Oct 28 2025

Status

approved