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A268758
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Number of polyominoes with width and height equal to 2n that are invariant under all symmetries of the square.
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3
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1, 3, 17, 163, 2753, 84731, 4879497, 535376723, 112921823249, 45931435159067, 36048888105745113, 54568015172025197171, 159197415409641803530753, 894444473815989281612355579, 9671160618112663336510127727593, 201110001346886305066013828873025811
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OFFSET
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1,2
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COMMENTS
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Also number of polyominoes with width and height equal to 2n - 1 that are invariant under all symmetries of the square.
Bisection of A268339.
The water retention model for mathematical surfaces is described in the link below. The definition of a "lake" in this model is related to a class of polyominoes in A268339. Percolation theory may refer to these structures as "clusters that touch all boundaries."
Transportation across the square lattice requires a path of continuous edge connected cells. Is a pattern that only connects two opposite boundaries of the square ranked differently from one that connects all four boundaries?
This sequence is part of a effort to classify water retention patterns in a square by their symmetry, their capacity to connect boundaries of the square and the number of edge cells that are connected across opposite boundaries.
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 1..20
Craig Knecht, Connections between boundaries of the square
Craig Knecht Retention capacity of a random surface, arXiv:1110.6166 [cond-mat.dis-nn], 2011-2012.
Wikipedia, Water Retention on Mathematical Surfaces
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FORMULA
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a(n) = A331878(n) - 3*A331878(n-1) + 3*A331878(n-2) - A331878(n-3) for n >= 4. - Andrew Howroyd, May 03 2020
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EXAMPLE
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For a(2) = 3: the three polyominoes of width and height 2*2 - 1 = 3 and the corresponding three polynomial of width and height 2*2 = 4 are shown below. Note that each even-dimension polyomino is produced by duplicating the center row/column of an odd-dimension polyomino.
3 X 3:
0 1 0 1 1 1 1 1 1
1 1 1 1 0 1 1 1 1
0 1 0 1 1 1 1 1 1
4 X 4:
0 1 1 0 1 1 1 1 1 1 1 1
1 1 1 1 1 0 0 1 1 1 1 1
1 1 1 1 1 0 0 1 1 1 1 1
0 1 1 0 1 1 1 1 1 1 1 1
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CROSSREFS
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Cf. A268339, A268404, A331878.
Sequence in context: A221410 A175607 A052143 * A069856 A214346 A015083
Adjacent sequences: A268755 A268756 A268757 * A268759 A268760 A268761
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KEYWORD
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nonn
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AUTHOR
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Craig Knecht, Feb 14 2016
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EXTENSIONS
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Terms a(9) and beyond from Andrew Howroyd, May 03 2020
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STATUS
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approved
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