A397023 Number of free polynars with n cells that are not the 2 X enlargement of a free n-omino.

0, 1, 7, 55, 455, 4191, 39864, 390534, 3884896, 39149945, 398346975, 4085997296

Offset

1,3

Comments

A390620(n) counts the free polynars with n cells, equivalently the n-fold multiples of the O-tetromino (edge-connected unions of n axis-aligned 2 X 2 blocks glued at integer offsets). Those whose blocks all lie on a common even sublattice are exactly the 2 X enlargements 2P of the free n-ominoes P, and there are A000105(n) of these; this sequence counts the rest. Equivalently, a(n) is the number of free polynars with n cells in which every tiling by 2 X 2 blocks uses at least one half-edge (one-step offset) gluing. See A397019 for the related family of tetromino multiples.

Links

Table of n, a(n) for n=1..12.

George Sicherman, Catalogue of Polynars.

Haoyang Xu, Illustration of a(2)

Index entries for sequences related to polyominoes.

Formula

a(n) = A390620(n) - A000105(n).

Example

a(2) = 1: realized as multiples of the O-tetromino, the two free polynars with 2 cells are two 2 X 2 blocks meeting along a full edge (the 2 X 4 rectangle, = 2 X the domino) and two blocks meeting along a half edge (one offset octomino); only the latter is not a 2 X enlargement, so a(2) = 1.
a(3) = A390620(3) - A000105(3) = 9 - 2 = 7.

Crossrefs

Cf. A390620 (free polynars), A000105 (free polyominoes), A390621 (one-sided polynars), A390622 (fixed polynars), A397019, A397020, A397021, A397022.

Sequence in context: A383602 A049028 A224274 * A096951 A113714 A246459

Adjacent sequences: A397020 A397021 A397022 * A397025 A397026 A397027

Keyword

nonn,more,new

Author

Haoyang Xu, Jun 19 2026

Status

approved