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A324406 Number of oriented polyomino rings of length 4n with fourfold rotational symmetry. 5

%I #9 Sep 30 2021 22:10:48

%S 1,1,1,2,4,8,16,33,67,141,295,630,1340,2895,6237,13596,29556,64846,

%T 141976,313131,689425,1527099,3377723,7508724,16671776,37175536,

%U 82809462,185141322,413555554,926743719,2075094083,4659549155,10455390287,23519366120,52872809784

%N Number of oriented 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. For oriented rings, chiral pairs (though congruent) are counted as two.

%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 Corrected; see A324408. - _Robert A. Russell_, Sep 30 2021

%H 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 a(n) = A324407(n) + A324408(n) = 2*A324407(n) - A324409(n) = 2*A324408(n) + A324409(n).

%e 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 Cf. A324407 (unoriented), A324408 (chiral), A324409 (achiral).

%Y Cf. A144553.

%K nonn,hard

%O 1,4

%A _Robert A. Russell_, Feb 26 2019

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