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A227004
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Irregular triangle read by rows: T(n,k) is the number of inequivalent tilings by squares of an n X n square lattice that contain k nodes unconnected to any of their neighbors.
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7
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1, 1, 1, 1, 1, 0, 0, 1, 1, 3, 4, 2, 2, 0, 0, 0, 0, 1, 1, 3, 13, 20, 17, 6, 10, 5, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 6, 37, 138, 280, 300, 255, 218, 98, 43, 55, 28, 20, 11, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1
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OFFSET
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1,10
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COMMENTS
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The n-th row contains (n-1)^2 + 1 elements.
The irregular triangle is shown below.
\ k 0 1 2 3 4 5 6 7 8 9 ...
n
1 1
2 1 1
3 1 1 0 0 1
4 1 3 4 2 2 0 0 0 0 1
5 1 3 13 20 17 6 10 5 0 1 ...
6 1 6 37 138 280 300 255 218 98 43 ...
7 1 6 75 505 2160 5410 8508 9179 8805 7917 ...
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LINKS
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FORMULA
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Sum_{k=0..(n-1)^2} T(n,k) = A224239(n).
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EXAMPLE
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For n = 4, there are 3 inequivalent tilings that contain 1 isolated node, so T(4,1) = 3.
A 2 X 2 square contains 1 isolated node.
Consider that each tiling is composed of ones and zeros where a one represents a node with one or more links to its neighbors and a zero represents a node with no links to its neighbors. Then the 3 tilings are:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 0 1 1 1 1 1 0 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 0 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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