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A121198 Number of one-sided chessboard polyominoes with n cells (similar to but different from A001071). 6
2, 1, 4, 10, 36, 110, 392, 1371, 5000, 18251, 67792, 253040, 952540, 3602846, 13699554, 52298057, 200406388, 770416390, 2970401696, 11482413680, 4449188109 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..21.

Joseph Myers, Chessboard polyominoes

Livio Zucca, PolyMultiForms

FORMULA

From John Mason, Jan 04 2014: (Start)

a(n) = 2*P(n) + 2*A(n) - R90C(n) - R90A(n) - R180S(n) - 2*R180A(n),

where P is the number of free polyominoes (A000105),

A is the number of free polyominoes lacking bilateral symmetry (A030228),

R90C is the number of free polyominoes of size n that have 90 degree rotational symmetry about a point that coincides with the corner of the grid, independently of any other symmetries (R90C is zero for polyominoes of size not a multiple of 4, as they cannot have the required symmetry - see A234009),

R90A is as R90C but only for those polyominoes lacking bilateral symmetry,

R180S is the number of free polyominoes of size 2n that have 180 degree rotational symmetry about a point that coincides with the mid-point of a side of a square of the grid, and that also have bilateral symmetry (R180S is zero for odd-sized polyominoes, as they cannot have the required symmetry), and

R180A is as R180S but for polyominoes lacking bilateral symmetry.  (End)

CROSSREFS

Cf. A001071, A001933, A121195, A121196, A121198.

Sequence in context: A198262 A085421 A001071 * A234599 A016544 A134028

Adjacent sequences:  A121195 A121196 A121197 * A121199 A121200 A121201

KEYWORD

nonn,more,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

STATUS

approved

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Last modified July 21 04:27 EDT 2017. Contains 289632 sequences.