

A324409


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


9



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
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OFFSET

1,4


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 onehalf a tile away from the center in both the horizontal and vertical directions, replacing those four centerline cells with four pairs of cells.


LINKS



FORMULA



EXAMPLE

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.


CROSSREFS



KEYWORD

nonn,hard


AUTHOR



STATUS

approved



