A224867 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 4 elements; triangle T(n,k), k >= 1, 0 <= n < k, read by columns.
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 21, 0, 0, 0, 10, 65, 440, 0, 0, 0, 27, 222, 1901, 14508, 0, 0, 0, 58, 676, 7716, 81119, 856559
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 0 0 0 0 0 0 ... 3 1 5 10 27 58 ... 4 21 65 222 676 ... 5 440 1901 7716 ... 6 14508 81119 ... 7 856559 ... ... T(3,5) = 5 because there are 5 different sets of 4 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 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 An example of a tiling in each set is: ._________. ._________. ._________. ._________. ._________. | |_|_|_| |_| |_|_| | | |_| | |_|_|_| | | | |_ _|_|_|_| |_|_ _|_|_| |_ _|_ _|_| |___| |_| |___| | |_|_|_|_|_| |_|_|_|_|_| |_|_|_|_|_| |_|_|___|_| |_|_|_____|
Links
- Christopher Hunt Gribble, C++ program