A347257 -id:A347257 - cp's OEIS frontend

A030225 Number of achiral hexagonal polyominoes with n cells.

1, 1, 3, 4, 11, 17, 46, 75, 202, 341, 914, 1581, 4222, 7436, 19794, 35357, 93859, 169558, 449039, 818793, 2163827, 3976636, 10489341, 19406704, 51103471, 95099113, 250040802, 467679257, 1227941119, 2307128946, 6049886572, 11412695367, 29891913576, 56593284153, 148067307799

Offset: 1

David W. Wilson

nonn

These are polyominoes of the Euclidean regular tiling of hexagons with Schläfli symbol {6,3}. This sequence can most readily be calculated by enumerating fixed polyominoes for three situations: 1) fixed polyominoes with a horizontal axis of symmetry along an edge of a cell with no cell centered on that axis, A001207(n/2), 2) fixed polyominoes with a horizontal axis of symmetry that is a diagonal of at least one cell, A347258, and 3) fixed polyominoes with a horizontal axis of symmetry that joins the midpoints of opposite edges of at least one cell, A347257. These three sequences include each achiral polyomino exactly twice. - Robert A. Russell, Aug 24 2021

Robert A. Russell, Table of n, a(n) for n = 1..36
Robert A. Russell, Examples for polyominoes with four or fewer cells

Cf. A006535 (oriented), A000228 (unoriented), A030226 (chiral).
Calculation components: A001207, A347257, A347258.
Other tilings: A030223 {3,6}, A030227 {4,4}.

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A000228 = A@000228;
A006535 = A@006535;
a[n_] := 2 A000228[[n]] - A006535[[n]];
a /@ Range[20] (* Jean-François Alcover, Feb 22 2020 *)

From Robert A. Russell, Aug 24 2021: (Start)
For odd n, a(n) = (A347257(n) + A347258(n)) / 2; for even n, a(n) = (A001207(n/2) + A347257(n) + A347258(n)) / 2.
a(n) = 2*A000228(n) - A006535(n) = A006535(n) - 2*A030226(n) = A000228(n) - A030226(n). (End)

More terms from Joseph Myers, Sep 21 2002

Name edited by Robert A. Russell, Aug 24 2021

A347258 Number of fixed hexagonal polyominoes with n cells that have a horizontal axis of symmetry that is a diagonal of at least one of the n cells.

Original entry on oeis.org

1, 0, 3, 1, 10, 5, 40, 23, 169, 107, 741, 499, 3334, 2349, 15278, 11141, 71012, 53198, 333756, 255553, 1582885, 1234059, 7563365, 5986757, 36367445, 29161696, 175810059, 142561190, 853868747, 699179932, 4163891024

Offset: 1

Robert A. Russell, Aug 24 2021

nonn

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 diagonal. 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.

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.

Cf. A001207, A347257, A030225.

cp's OEIS Frontend

A030225 Number of achiral hexagonal polyominoes with n cells.

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