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
KEYWORD
nonn,tabf
AUTHOR
Christopher Hunt Gribble, Jul 26 2013
STATUS
approved