A324408 Number of chiral pairs of polyomino rings of length 4n with fourfold rotational symmetry.

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

Author

Robert A. Russell, Feb 26 2019

Keywords

• nonn
• hard

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

Examples

			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
		

Links

• D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.

Crossrefs

Cf. A324406 (oriented), A324407 (unoriented), A324409 (achiral).

Cf. also A144553.

Formula

a(n) = A324406(n) - A324407(n) = (A324406(n) - A324409(n)) / 2 = A324407(n) - A324409(n).