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%I #30 Sep 06 2021 05:11:56
%S 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,
%T 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,
%U 0,0,0,0,0,0,1
%N 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.
%C The number of entries per row is given by A225568.
%H Christopher Hunt Gribble, <a href="/A225777/b225777.txt">Rows 1..36 for n=2..8 and k=1..n flattened</a>
%H Christopher Hunt Gribble, <a href="/A225777/a225777.cpp.txt">C++ program</a>
%F 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.
%F Sum_{u=1..(n-1)^2} T(n,n,u) = A045846(n).
%e The irregular triangle begins:
%e n,k\u 0 1 2 3 4 5 6 7 8 9 10 11 12 ...
%e 1,1 1
%e 2,1 1
%e 2,2 1 1
%e 3,1 1
%e 3,2 1 2
%e 3,3 1 4 0 0 1
%e 4,1 1
%e 4,2 1 3 1
%e 4,3 1 6 4 0 2
%e 4,4 1 9 16 8 5 0 0 0 0 1
%e 5,1 1
%e 5,2 1 4 3
%e 5,3 1 8 12 0 3 4
%e 5,4 1 12 37 34 15 12 4 0 0 2
%e 5,5 1 16 78 140 88 44 68 32 0 4 0 0 0 ...
%e ...
%e 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:
%e 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 1
%e 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e 1 0 1 1 1 1 0 1 1 0 1 1 1 1 0 1
%e 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%Y Cf. A045846, A219924, A226997, A225542, A225803, A225568.
%K nonn,tabf
%O 1,7
%A _Christopher Hunt Gribble_, Jul 26 2013
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