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A324406 Number of oriented polyomino rings of length 4n with fourfold rotational symmetry. 5
1, 1, 1, 2, 4, 8, 16, 33, 67, 141, 295, 630, 1340, 2895, 6237, 13596, 29556, 64846, 141976, 313131, 689425, 1527099, 3377723, 7508724, 16671776, 37175536, 82809462, 185141322, 413555554, 926743719, 2075094083, 4659549155, 10455390287, 23519366120, 52872809784 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
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.
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.
Corrected; see A324408. - Robert A. Russell, Sep 30 2021
LINKS
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
FORMULA
a(n) = A324407(n) + A324408(n) = 2*A324407(n) - A324409(n) = 2*A324408(n) + A324409(n).
EXAMPLE
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.
CROSSREFS
Cf. A324407 (unoriented), A324408 (chiral), A324409 (achiral).
Cf. A144553.
Sequence in context: A119610 A121485 A308808 * A182442 A098588 A367715
KEYWORD
nonn,hard
AUTHOR
Robert A. Russell, Feb 26 2019
STATUS
approved

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Last modified June 17 13:47 EDT 2024. Contains 373445 sequences. (Running on oeis4.)