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

Author

N. J. A. Sloane, Aug 17 2006

Keywords

• nonn
• hard

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

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).

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)

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