A030227 Number of achiral polyominoes with n cells.

1, 1, 1, 2, 3, 6, 10, 20, 34, 70, 121, 250, 441, 912, 1630, 3375, 6092, 12624, 22961, 47616, 87136, 180811, 332549, 690398, 1275166, 2648422, 4909364, 10199792, 18966700, 39416488, 73497642, 152777230, 285569898, 593717419, 1112188817, 2312672439, 4340728280

Offset

0,4

Comments

Polyominoes with n cells and at least one line of reflection symmetry. - Joshua Zucker, Mar 08 2008

This sequence can most readily be calculated by enumerating fixed polyominoes for three different axes of symmetry: 1) a line composed of the diagonals of cells, A346800, 2) a line composed of edges of cells, and 3) a line composed of lines connecting the centers of adjacent cells, A346799. For the second case, any fixed polyomino just touching the edge line is reflected on the other side, so that sequence is A001168(n/2) for even values of n and zero otherwise. These three sequences together include each achiral polyomino exactly twice. - Robert A. Russell, Aug 04 2021

Links

John Mason, Table of n, a(n) for n = 0..50 (terms 1..48 from Robert A. Russell.)

D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.

Toshihiro Shirakawa, Enumeration of Polyominoes considering the symmetry, April 2012, pp. 3-4.

Formula

a(n) = A000105(n) - A030228(n) = 2*A000105(n) - A000988(n). - Andrew Howroyd, Dec 04 2018

a(n) = A006746(n) + A006748(n) + A056877(n) + A056878(n) + A142886(n) = A000988(n) - 2*A030228(n). - Robert A. Russell, Feb 02 2019

For odd n, a(n) = (A346799(n) + A346800(n)) / 2; for even n, a(n) = (A346799(n) + A001168(n/2) + A346800(n)) / 2. - Robert A. Russell, Aug 04 2021

Example

For a(4)=3, the achiral tetrominoes are a 2 X 2 square, a 1 X 4 rectangle, and a cell plus three cells adjacent to it (forming a shortened T).

Mathematica

A000105 = Cases[Import["https://oeis.org/A000105/b000105.txt", "Table"], {_, _}][[All, 2]];
A000988 = Cases[Import["https://oeis.org/A000988/b000988.txt", "Table"], {_, _}][[All, 2]];
a[n_] := 2*A000105[[n + 1]] - A000988[[n]];
Array[a, 45] (* Jean-François Alcover, Sep 08 2019, after Andrew Howroyd *)

Crossrefs

Cf. A000988 (oriented), A000105 (unoriented), A030228 (chiral).

Cf. A006746, A006748, A056877, A056878, A142886 (subcategories).

Cf. A001168, A346799, A346800.

Sequence in context: A135452 A077027 A030436 * A180272 A319436 A061551

Adjacent sequences: A030224 A030225 A030226 * A030228 A030229 A030230

Keyword

nonn

Author

David W. Wilson

Extensions

a(23)-a(36) from Andrew Howroyd, Dec 04 2018

Name edited by Robert A. Russell, Feb 03 2019

Offset changed to 0, and a(0) added by John Mason, Jan 12 2023

Status

approved