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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 (list; graph; refs; listen; history; text; internal format)
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

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 90-degree 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 180-degree rotational symmetry about a point that coincides with the mid-side of a square. Note that for polyominoes which have a hole in the center, the center of rotation will be the mid-side 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

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Last modified August 21 00:45 EDT 2017. Contains 290855 sequences.