A346800 - OEIS

A346800

Number of fixed polyominoes with n cells that have a diagonal axis of symmetry going from lower left to upper right.

1, 0, 2, 1, 5, 4, 16, 13, 54, 46, 186, 167, 660, 612, 2384, 2267, 8726, 8464, 32278, 31822, 120419, 120338, 452420, 457320, 1709845, 1745438, 6494848, 6686929, 24779026, 25703792, 94899470, 99096382, 364680344

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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 DL sequence in the Shirakawa link. The sequence can be calculated using Redelmeier's method; one chooses an original cell such that no cells in its LL-UR diagonal on one side of it are eligible, nor are any cells in lower LL-UR diagonals. Cells in that original diagonal are counted as one; all others count as two. Jensen's transfer matrix method (see Knuth POLYNUM program) could likely be modified to enumerate this sequence for many more terms; instead of rows, one uses diagonals.

The sequence also enumerates free polyominoes of size 4*n with maximal symmetry that have a center of rotation on a vertex of the underlying square matrix, which are a subset of those enumerated by A142886. - John Mason Jan 27 2022

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) = 2*A006748(n) + 2*A056878(n) + A142886(n). - John Mason Jan 27 2022

EXAMPLE

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

X X XX XX XXX

X XXX XX X X

CROSSREFS