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A225777
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Number T(n,k,u) of distinct tilings of an n X k rectangle using integer-sided square tiles containing u nodes that are unconnected to any of their neighbors; irregular triangle T(n,k,u), 1 <= k <= n, u >= 0, read by rows.
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5
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1, 1, 1, 1, 1, 1, 2, 1, 4, 0, 0, 1, 1, 1, 3, 1, 1, 6, 4, 0, 2, 1, 9, 16, 8, 5, 0, 0, 0, 0, 1, 1, 1, 4, 3, 1, 8, 12, 0, 3, 4, 1, 12, 37, 34, 15, 12, 4, 0, 0, 2, 1, 16, 78, 140, 88, 44, 68, 32, 0, 4, 0, 0, 0, 0, 0, 0, 1
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OFFSET
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1,7
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COMMENTS
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The number of entries per row is given by A225568.
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LINKS
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FORMULA
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T(n,k,0) = 1, T(n,k,1) = (n-1)(k-1), T(n,k,2) = (n^2(k-1) - n(2k^2+5k-13) + (k^2+13k-24))/2.
Sum_{u=1..(n-1)^2} T(n,n,u) = A045846(n).
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EXAMPLE
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The irregular triangle begins:
n,k\u 0 1 2 3 4 5 6 7 8 9 10 11 12 ...
1,1 1
2,1 1
2,2 1 1
3,1 1
3,2 1 2
3,3 1 4 0 0 1
4,1 1
4,2 1 3 1
4,3 1 6 4 0 2
4,4 1 9 16 8 5 0 0 0 0 1
5,1 1
5,2 1 4 3
5,3 1 8 12 0 3 4
5,4 1 12 37 34 15 12 4 0 0 2
5,5 1 16 78 140 88 44 68 32 0 4 0 0 0 ...
...
For n = 4, k = 3, there are 4 tilings that contain 2 isolated nodes, so T(4,3,2) = 4. 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 4 tilings are:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 0 1 1 1 1 0 1 1 0 1 1 1 1 0 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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