A224861 Number T(n,k) of tilings of an n X k rectangle using integer-sided square tiles reduced for symmetry, where the orbits under the symmetry group of the rectangle, D2, have 2 elements; triangle T(n,k), k >= 1, 0 <= n < k, read by columns.
0, 0, 0, 0, 0, 1, 0, 0, 1, 4, 0, 0, 3, 3, 15, 0, 0, 4, 9, 38, 75, 0, 0, 9, 9, 68, 77, 604, 0, 0, 13, 21, 160, 311, 2384, 4556
Offset: 1
Examples
The triangle is: n\k 1 2 3 4 5 6 7 8 ... . 0 0 0 0 0 0 0 0 0 ... 1 0 0 0 0 0 0 0 ... 2 1 1 3 4 9 13 ... 3 4 3 9 9 21 ... 4 15 38 68 160 ... 5 75 77 311 ... 6 604 2384 ... 7 4556 ... ... T(3,5) = 3 because there are 3 different sets of 2 tilings of the 3 X 5 rectangle by integer-sided squares, where any sequence of group D2 operations will transform each tiling in a set into the other 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 An example of a tiling in each set is: ._________. ._________. ._________. | |_| | | |_|_|_| | |_|_| |_ _|_|_ _| |___|_| | | |_|_| |_|_|_|_|_| |_|_|_|___| |_____|_|_|
Links
- Christopher Hunt Gribble, C++ program