cp's OEIS Frontend

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

%I A324407 #8 Sep 30 2021 22:10:57
%S A324407 1,1,1,2,3,6,10,21,38,80,157,336,691,1493,3164,6900,14880,32647,71212,
%T A324407 157069,345216,764666,1689978,3756879,8338405,18593389,41410352,
%U A324407 92583361,206790477,463400376,1037575558,2329839141,5227759707,11759828568,26436550400
%N A324407 Number of unoriented polyomino rings of length 4n with fourfold rotational symmetry.
%C A324407 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.
%C A324407 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 A324407 Corrected; see A324408. - _Robert A. Russell_, Sep 30 2021
%H A324407 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 A324407 a(n) = A324406(n) - A324408(n) = (A324406(n) + A324409(n)) / 2 = A324408(n) + A324409(n).
%e A324407 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.
%Y A324407 Cf. A324406 (oriented), A324408 (chiral), A324409 (achiral).
%Y A324407 Cf. A144553.
%K A324407 nonn,hard
%O A324407 1,4
%A A324407 _Robert A. Russell_, Feb 26 2019