%I #17 Feb 01 2021 02:18:15
%S 1,1,1,2,2,4,4,9,9,19,19,42,42,91,91,204,204,448,448,1007,1007,2233,
%T 2233,5034,5034,11242,11242,25400,25400,57033,57033,129127,129127,
%U 291016,291016
%N Number of achiral polyomino rings of length 4n with fourfold rotational symmetry.
%C 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.
%C 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.
%C 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.
%H D. H. Redelmeier, <a href="http://dx.doi.org/10.1016/0012365X(81)902375">Counting polyominoes: yet another attack</a>, Discrete Math., 36 (1981), 191203.
%F a(n) = 2*A324407(n)  A324406(n) = A324406(n)  2*A324408(n)) / 2 = A324407(n)  A324408(n).
%e For a(1)=1, the four cells form a square.
%e For a(2)=1, the eight cells form a 3 X 3 square with the center cell omitted.
%e For a(3)=1, the twelve cells form a 4 X 4 square with the four inner cells omitted.
%e 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.
%Y Cf. A324406 (oriented), A324407 (unoriented), A324408 (chiral).
%Y Cf. A144553.
%K nonn,hard
%O 1,4
%A _Robert A. Russell_, Feb 26 2019
