A001933 Number of chessboard polyominoes with n squares. (Formerly M0171 N0066)

2, 1, 4, 7, 24, 62, 216, 710, 2570, 9215, 34146, 126853, 477182, 1802673, 6853152, 26153758, 100215818, 385226201, 1485248464, 5741275753, 22246121356, 86383454582, 336094015456, 1309998396933, 5114454089528, 19998173763831, 78306021876974, 307022186132259, 1205243906123956, 4736694016531135

Offset

1,1

Comments

Chessboard-colored polyominoes, considering to be distinct two shapes that cannot be mapped onto each other by any form of symmetry. For example, there are two distinct monominoes, one black, one white. There is only one domino, with one black square, and one white. - John Mason, Nov 25 2013

References

W. F. Lunnon, personal communication.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

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

Joseph Myers, Chessboard polyominoes

Formula

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

For n multiple of 2 but not of 4, a(n) = 2*A000105(n) - (A234006(n/2) + A234008(n/2)).

For n multiple of 4, a(n) = 2*A000105(n) - (A234006(n/2) + A234008(n/2) + A234007(n/4)). - John Mason, Dec 23 2021

Crossrefs

Cf. A001071, A000105, A121198, A234006 (free polyominoes of size 2n that have at least reflectional symmetry on a horizontal or vertical axis that coincides with the edges of some of the squares), A234007 (free polyominoes with 4n squares, having 90-degree rotational symmetry about a square corner, but not having reflective symmetry), A234008 (free polyominoes with 2n squares, having 180-degree rotational symmetry about a square mid-side, but no reflective symmetry).

Sequence in context: A045625 A294501 A146004 * A038557 A348678 A011234

Adjacent sequences: A001930 A001931 A001932 * A001934 A001935 A001936

Keyword

hard,nonn

Author

N. J. A. Sloane

Extensions

a(14)-a(17) from Joseph Myers, Oct 01 2011

a(18)-a(23) from John Mason, Dec 05 2013

a(24)-a(30) from John Mason, Dec 23 2021

Status

approved