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 A225777 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. 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS The number of entries per row is given by A225568. LINKS Christopher Hunt Gribble, Rows 1..36 for n=2..8 and k=1..n flattened Christopher Hunt Gribble, C++ program FORMULA 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). EXAMPLE 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 CROSSREFS Cf. A045846, A219924, A226997, A225542, A225803, A225568. Sequence in context: A096097 A212805 A246431 * A016585 A309054 A143316 Adjacent sequences: A225774 A225775 A225776 * A225778 A225779 A225780 KEYWORD nonn,tabf AUTHOR Christopher Hunt Gribble, Jul 26 2013 STATUS approved

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Last modified September 12 09:10 EDT 2024. Contains 375850 sequences. (Running on oeis4.)