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A268311
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Number of free polyominoes that form a continuous path of edge joined cells spanning an n X n square in both dimensions.
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10
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OFFSET
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1,2
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COMMENTS
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This idea originated from the water retention model for mathematical surfaces and is identical to the concept of a "lake". A lake is body of water that has dimensions of (n-2) X (n-2) when the square size is n X n. All other bodies of water are "ponds".
Iwan Jensen with his transfer matrix algorithm provided the number of symmetrically redundant solutions. Walter Trump enumerated the symmetrically unique solutions.
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LINKS
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R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy) page 9 (incorrect at n=15).
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EXAMPLE
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The cells with value 1 show the smallest possible lake in this 4 X 4 square:
1 1 1 1
0 0 0 1
0 0 0 1
0 0 0 1
a(3)=24 = 6+7+7+3+1: There fit 6 5-ominoes in a 3x3 square, 7 6-ominoes in a 3x3 square, 7 7-ominoes in a 3x3 square, 3 8-ominoes in a 3x3 square, a 1 9-omino in a 3x3 square. - R. J. Mathar, Jun 07 2020
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CROSSREFS
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Cf. A054247 (all unique water retention patterns). Diagonal of A268371.
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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