A225803 Number T(n,k,u) of tilings of an n X k rectangle using integer-sided square tiles, reduced for symmetry, 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.
1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 0, 1, 1, 1, 2, 2, 1, 2, 4, 0, 2, 1, 1, 4, 13, 10, 6, 3, 1, 0, 0, 1, 1, 1, 3, 4, 1, 1, 3, 8, 3, 2, 3, 0, 0, 1, 1, 6, 23, 33, 24, 15, 6, 0, 2, 2, 2, 1, 1, 6, 40, 101, 129, 79, 74, 53, 13, 9, 11, 4, 0, 0, 0, 0, 1
Offset: 1
Examples
The irregular triangle T(n,k,u) begins: n,k\u 0 1 2 3 4 5 6 7 8 9 10 11 12 ... 2,1 1 3,1 1 3,2 1 1 4,1 1 4,2 1 2 1 4,3 1 2 2 0 1 5,1 1 5,2 1 2 2 5,3 1 2 4 0 2 1 5,4 1 4 13 10 6 3 1 0 0 1 6,1 1 6,2 1 3 4 1 6,3 1 3 8 3 2 3 0 0 1 6,4 1 6 23 33 24 15 6 0 2 2 1 6,5 1 6 40 101 79 74 53 13 9 11 4 0 0 ... ... T(5,3,2) = 4 because there are 4 different sets of tilings of the 5 X 3 rectangle by integer-sided squares in which each tiling contains 2 isolated nodes. Any sequence of group D2 operations will transform each tiling in a set into another in the same set. Group D2 operations are: . the identity operation . rotation by 180 degrees . reflection about a horizontal axis through the center . reflection about a vertical axis through the center 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. An example of a tiling in each set is: 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 0 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 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Links
- Christopher Hunt Gribble, Rows 1..28 for n = 2..8 and k = 1..n-1 flattened
- Christopher Hunt Gribble, C++ program
Formula
T1(n,k,0) = 1, T1(n,k,1) = floor(n/2)*floor(k/2).
Comments