cp's OEIS Frontend

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

%I A324409 #21 Dec 15 2024 15:35:51
%S A324409 1,1,1,2,2,4,4,9,9,19,19,42,42,91,91,204,204,448,448,1007,1007,2233,
%T A324409 2233,5034,5034,11242,11242,25400,25400,57033,57033,129127,129127,
%U A324409 291016,291016
%N A324409 Number of achiral polyomino rings of length 4n with fourfold rotational symmetry.
%C A324409 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 A324409 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 A324409 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.
%H A324409 D. H. Redelmeier, <a href="http://dx.doi.org/10.1016/0012-365X(81)90237-5">Counting polyominoes: yet another attack</a>, Discrete Math., 36 (1981), 191-203.
%F A324409 a(n) = 2*A324407(n) - A324406(n) = A324406(n) - 2*A324408(n) = A324407(n) - A324408(n).
%e A324409 For a(1)=1, the four cells form a square.
%e A324409 For a(2)=1, the eight cells form a 3 X 3 square with the center cell omitted.
%e A324409 For a(3)=1, the twelve cells form a 4 X 4 square with the four inner cells omitted.
%e A324409 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 A324409 Cf. A324406 (oriented), A324407 (unoriented), A324408 (chiral).
%Y A324409 Cf. A144553.
%K A324409 nonn,hard
%O A324409 1,4
%A A324409 _Robert A. Russell_, Feb 26 2019