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A324408
Number of chiral pairs of polyomino rings of length 4n with fourfold rotational symmetry.
9
0, 0, 0, 0, 1, 2, 6, 12, 29, 61, 138, 294, 649, 1402, 3073, 6696, 14676, 32199, 70764, 156062, 344209, 762433, 1687745, 3751845, 8333371, 18582147, 41399110, 92557961, 206765077, 463343343, 1037518525, 2329710014, 5227630580, 11759537552, 26436259384
OFFSET
1,6
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. Each chiral ring is congruent to but different from its reflection; the two form a chiral pair.
These chiral rings have fourfold 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.
In early September, 2021, John Mason informed me that a(16) should be 6696 instead of 6695. He supplied me with representations of all of the rings, and I slowly discovered that my program had missed one and had serious errors. After I corrected it, we did match new values for a(16), a(18), a(20), and a(22). We are reasonably confident that the values shown are now correct. - Robert A. Russell, Sep 30 2021
LINKS
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
FORMULA
a(n) = A324406(n) - A324407(n) = (A324406(n) - A324409(n)) / 2 = A324407(n) - A324409(n).
EXAMPLE
For a(5) = 1, the pair is XXX XXX .
X XXX XXX X
XX X X XX
X XX XX X
XXX X X XXX
XXX XXX
CROSSREFS
Cf. A324406 (oriented), A324407 (unoriented), A324409 (achiral).
Cf. also A144553.
Sequence in context: A057582 A094779 A093387 * A229487 A195166 A225646
KEYWORD
nonn,hard
AUTHOR
Robert A. Russell, Feb 26 2019
STATUS
approved