A324407 Number of unoriented polyomino rings of length 4n with fourfold rotational symmetry.

1, 1, 1, 2, 3, 6, 10, 21, 38, 80, 157, 336, 691, 1493, 3164, 6900, 14880, 32647, 71212, 157069, 345216, 764666, 1689978, 3756879, 8338405, 18593389, 41410352, 92583361, 206790477, 463400376, 1037575558, 2329839141, 5227759707, 11759828568, 26436550400

Offset: 1

Author

Robert A. Russell, Feb 26 2019

Keywords

• nonn
• hard

Comments

Redelmeier uses these rings to enumerate polyominoes of the regular tiling {4,4} with fourfold rotational symmetry (A144553) and an even number of cells. Each cell of a polyomino ring is adjacent to (shares an edge with) exactly two other cells. For unoriented rings, a chiral ring and its congruent reflection are counted as one.

For n odd, the center of the ring is a vertex of the tiling; for n even, the center is the center of a tile.

Corrected; see A324408. - Robert A. Russell, Sep 30 2021

Examples

			For a(1)=1, the four cells form a square. For a(2)=1, the eight cells form a 3 X 3 square with the center cell omitted. For a(3)=1, the twelve cells form a 4 X 4 square with the four inner cells omitted. For a(4)=2, the sixteen cells of one ring form a 5 X 5 square with the nine inner cells omitted; the other ring is similar, but with each corner cell omitted and replaced with the cell diagonally toward the center from that corner cell.
		

Links

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

Crossrefs

Cf. A324406 (oriented), A324408 (chiral), A324409 (achiral).

Cf. A144553.

Formula

a(n) = A324406(n) - A324408(n) = (A324406(n) + A324409(n)) / 2 = A324408(n) + A324409(n).