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

Author

David W. Wilson

Keywords

• nonn

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

Examples

			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).
		

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.

Crossrefs

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

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

Cf. A001168, A346799, A346800.

Programs

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 *)

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

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