A121198 Number of one-sided chessboard polyominoes with n cells (similar to but different from A001071).

2, 1, 4, 10, 36, 110, 392, 1371, 5000, 18251, 67792, 253040, 952540, 3602846, 13699554, 52298057, 200406388, 770416390, 2970401696, 11482413680, 44491881090, 172766379334, 672186650116, 2619994749395, 10228902882212, 39996339612824, 156612023354364, 614044341535992

Offset

1,1

Comments

Consider the tiling of the plane with squares of two different sizes as seen for example in Fig. 2.4.2(g) of Grünbaum and Shephard, p. 74. Sequence gives the number of "n-PairSquares", that is, polyominoes or animals that can be formed on this tiling from "n big or little squares, where the conjunction between two squares must involve an entire edge at least". - Original description (N. J. A. Sloane, Aug 17 2006, with quote from Livio Zucca's site)

Also counts one-sided polyominoes cut from an infinite chessboard with the usual coloring (big and little squares in Fig. 2.4.2(g) of Grünbaum and Shephard are equivalent to the two colors on a chessboard, and ignoring connections that are not a whole edge of one square means the connectivity is also equivalent); see Myers link regarding difference from A001071 for even terms a(6) onwards. - Joseph Myers, Oct 01 2011

References

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987.

Links

John Mason, Table of n, a(n) for n = 1..50

Joseph Myers, Chessboard polyominoes

Livio Zucca, PolyMultiForms

Formula

From John Mason, Dec 24 2021: (Start)

For odd n, a(n) = 2*A000105(n) + 2*A030228(n).

For n multiple of 2 but not of 4, a(n) = 2*A000105(n) + 2*A030228(n) - A346799(n/2) - 2*A234008(n/2).

For n multiple of 4, a(n) = 2*A000105(n) + 2*A030228(n) - A346799(n/2) - 2*A234008(n/2) - A234009(n/4) - A234007(n/4). (End)

Crossrefs

Cf. A001071, A001933, A121195, A121196, A000105 (free polyominoes), A030228 (chiral polyominoes), A234009 (free polyominoes with 90-degree rotational symmetry about a square corner), A234007 (chiral polyominoes with 90-degree rotational symmetry about a square corner), A346799 (achiral polyominoes with twofold rotational symmetry around the center of an edge), A234008 (chiral polyominoes with 180-degree rotational symmetry about the center of an edge).

Sequence in context: A198262 A085421 A001071 * A234599 A016544 A134028

Adjacent sequences: A121195 A121196 A121197 * A121199 A121200 A121201

Keyword

nonn,hard

Author

N. J. A. Sloane, Aug 17 2006

Extensions

a(6)-a(17) by Joseph Myers, Oct 01 2011

a(18)-a(21) by John Mason, Jan 04 2014

Erroneous a(21) removed by John Mason, Feb 12 2021

a(21)-a(28) from John Mason, Dec 24 2021

Status

approved