A324409 Number of achiral polyomino rings of length 4n with fourfold rotational symmetry.

1, 1, 1, 2, 2, 4, 4, 9, 9, 19, 19, 42, 42, 91, 91, 204, 204, 448, 448, 1007, 1007, 2233, 2233, 5034, 5034, 11242, 11242, 25400, 25400, 57033, 57033, 129127, 129127, 291016, 291016

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. Each achiral ring is identical to its reflection and has eightfold symmetry.

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.

For k > 0, the numbers of achiral rings with 8k and 8k+4 cells are the same. In the former, there are four cells in the same row or column as the center tile; we obtain the latter by moving all the cells one-half a tile away from the center in both the horizontal and vertical directions, replacing those four center-line cells with four pairs of cells.

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), A324407 (unoriented), A324408 (chiral).

Cf. A144553.

Formula

a(n) = 2*A324407(n) - A324406(n) = A324406(n) - 2*A324408(n) = A324407(n) - A324408(n).