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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 (list; graph; refs; listen; history; text; internal format)
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 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.
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
a(n) = 2*A324407(n) - A324406(n) = A324406(n) - 2*A324408(n)) / 2 = A324407(n) - A324408(n).
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.
Cf. A324406 (oriented), A324407 (unoriented), A324408 (chiral).
Cf. A144553.
Sequence in context: A292941 A074818 A322112 * A110199 A358429 A222736
Robert A. Russell, Feb 26 2019

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Last modified July 24 02:45 EDT 2024. Contains 374575 sequences. (Running on oeis4.)