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

1,3

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

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.

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

Example

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

Crossrefs

Cf. A030227, A056877, A346800.

Sequence in context: A263402 A047082 A062113 * A130968 A007748 A126617

Adjacent sequences: A346796 A346797 A346798 * A346800 A346801 A346802

Keyword

nonn

Author

Robert A. Russell, Aug 04 2021

Status

approved