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

Offset

1,3

Comments

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

Links

Robert A. Russell, Table of n, a(n) for n = 1..36

Robert A. Russell, Examples for polyominoes with four or fewer cells

Formula

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)

Mathematica

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 *)

Crossrefs

Cf. A006535 (oriented), A000228 (unoriented), A030226 (chiral).

Calculation components: A001207, A347257, A347258.

Other tilings: A030223 {3,6}, A030227 {4,4}.

Sequence in context: A026753 A027222 A026380 * A339157 A060285 A025079

Adjacent sequences: A030222 A030223 A030224 * A030226 A030227 A030228

Keyword

nonn,more

Author

David W. Wilson

Extensions

More terms from Joseph Myers, Sep 21 2002

Name edited by Robert A. Russell, Aug 24 2021

Status

approved