A346799 Number of fixed polyominoes with n cells that have a horizontal axis of symmetry that passes through the centers of cells.

1, 1, 2, 3, 7, 10, 24, 36, 86, 133, 314, 499, 1164, 1888, 4366, 7192, 16522, 27548, 62954, 106004, 241203, 409492, 928376, 1587151, 3586999, 6169400, 13904736, 24041597, 54053950, 93896826, 210654990, 367450477, 822754494

Offset: 1

Author

Robert A. Russell, Aug 04 2021

Keywords

• nonn

Comments

This is one of three sequences needed to calculate the number of achiral polyominoes, A030227. The three sequences together contain exactly two copies of each achiral polyomino. This is the FL sequence in the Shirakawa link. The sequence can be quickly calculated using Redelmeier's method; each polyomino cell in the lowest row is counted as one, while all the other polyomino cells are counted as two. Jensen's transfer matrix method (see Knuth POLYNUM program) could be modified to enumerate this sequence for over 100 terms; one needs only to keep track of the number of polyomino cells in the original row.

John Mason has pointed out that a(n) is also the number of achiral (2n)-ominoes with twofold rotational symmetry centered at the center of an edge. Just add to each polyomino its reflection in its leftmost edge to obtain these, the subset of A056877 with edge centers. - Robert A. Russell, Dec 15 2021

Examples

			For a(5)=7, the polyominoes are:    X
X       X   XX   XX    X            X
XXX   XXX   X     X   XXX   XXXXX   X
X       X   XX   XX    X            X
                                    X
		

Links

• John Mason, Table of n, a(n) for n = 1..50 (terms 1..47 from Robert A. Russell).
• D. E. Knuth, Program
• 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. A030227, A056877, A346800.

Formula

a(n) = A351127(n) + 2 * A351190(n) + A346799(n / 2) + 2 * A349328(n), setting A346799(n / 2) = 0 for noninteger arguments. - John Mason, Mar 13 2023