

A001933


Chessboard polyominoes with n squares.
(Formerly M0171 N0066)


10



2, 1, 4, 7, 24, 62, 216, 710, 2570, 9215, 34146, 126853, 477182, 1802673, 6853152, 26153758, 100215818, 385226201, 1485248464, 5741275753, 22246121356, 86383454582, 336094015456
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OFFSET

1,1


COMMENTS

Chessboardcolored 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

Table of n, a(n) for n=1..23.
Joseph Myers, Chessboard polyominoes


FORMULA

For odd n, a(n) = 2*A000105(n).  John Mason, Nov 27 2013
For even n, a(n) = 2*A000105(n)  (M(n) + R90(n) + R180(n))
Where:
M(n) is the number of free polyominoes of size n that have reflectional symmetry on a horizontal or vertical axis that coincides with the edges of some of the squares. (Note for example that the 3 X 3 square nonomino is not included, as the axes of symmetry do not coincide with the edges of any squares.) M(n) = 0 for all n not multiples of 2.
R90(n) is the number of free polyominoes of size n that have 90degree rotational symmetry about a point that coincides with the corner of a square. Note that for polyominoes which have a hole in the center, the center of rotation will be the corner of a square within the hole, rather than being the corner of a square of the polyomino itself. R90(n) = 0 for all n not multiples of 4. Exclude from R90(n) any polyominoes already in M(n).
R180 (n) is the number of free polyominoes of size n that have 180degree rotational symmetry about a point that coincides with the midside of a square. Note that for polyominoes which have a hole in the center, the center of rotation will be the midside of a square within the hole, rather than being within a square of the polyomino itself. R180 (n) = 0 for all n not multiples of 2. Exclude from R180 (n) any polyominoes already in M(n).
 John Mason, Dec 05 2013


CROSSREFS

Cf. A001071, A000105, A121198.
Sequence in context: A184345 A045625 A146004 * A038557 A011234 A208917
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


STATUS

approved



