A234008 Free polyominoes with 2n squares, having 180-degree rotational symmetry about a square mid-side, but no reflective symmetry.

0, 1, 4, 16, 60, 231, 877, 3362, 12905, 49825, 193003, 750581, 2927792, 11453171, 44911853, 176499605, 694954416, 2741031257, 10827727980, 42831355495, 169640762209, 672657218163, 2669990735153, 10608176066076, 42184579054003, 167887980491146, 668670899820289, 2665072331352491, 10628905413277882

Offset

1,3

Comments

The number of free polyominoes of size 2n that have 180-degree rotational symmetry about a point that coincides with the midpoint of a side a square, and that have not at the same time any reflective symmetry. Note that for polyominoes which have a hole in the center, the center of rotation will be the midpoint of a side of a square within the hole, rather than being the midpoint of a side of a square of the polyomino itself. The sequence is defined for 2n rather than n as odd-sized polyominoes cannot have the required symmetry.

The sequence enumerates a subset of the polyominoes enumerated by A006747.

Links

Table of n, a(n) for n=1..29.

Toshihiro Shirakawa, Enumeration of Polyominoes up to Size N=59, arXiv:2510.22446 [math.CO], 2025. See p. 5-6.

Formula

a(n) = (A390654(n) - A346799(n))/2. - John Mason, Nov 14 2025

Crossrefs

Cf. A000105, A001933, A234006, A234007, A234009, A234010, A006747, A390654, A346799.

Sequence in context: A047097 A051043 A123620 * A355351 A203153 A380404

Adjacent sequences: A234005 A234006 A234007 * A234009 A234010 A234011

Keyword

nonn,hard

Author

John Mason, Dec 18 2013

Extensions

a(12)-a(18) from John Mason, Dec 13 2021

a(19)-a(25) from John Mason, Apr 15 2023

a(26)-a(29) from John Mason, Nov 14 2025

Status

approved