A347257 Number of fixed hexagonal polyominoes with n cells that have a horizontal axis of symmetry that connects the midpoints of opposite edges of one of the n cells.

1, 1, 3, 4, 12, 18, 52, 83, 235, 389, 1087, 1849, 5110, 8871, 24310, 42884, 116706, 208559, 564322, 1019362, 2744769, 5003180, 13415317, 24644438, 65839497, 121769444, 324271545, 603304529, 1602013491, 2996240586

Offset: 1

Author

Robert A. Russell, Aug 24 2021

Keywords

• nonn

Comments

These are polyominoes of the Euclidean hexagonal regular tiling with Schläfli symbol {6,3}. This is one of three sequences needed to calculate the number of achiral polyominoes, A030225. The three sequences together contain exactly two copies of each achiral polyomino. This sequence can be calculated using a modification of Redelmeier's method; one chooses an original cell that is leftmost on and bisected by the axis of symmetry along a horizontal line connecting midpoints of opposite edges of one cell. Neighbors are added only if their centers are above the axis of symmetry or on the axis of symmetry to the right of the original cell. Cells not centered on the axis of symmetry are counted twice to include their reflections.

Links

• Robert A. Russell, Table of n, a(n) for n = 1..36
• D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.

Crossrefs

Cf. A001207, A347258, A030225.