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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.
4
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
1,14
LINKS
Christopher Hunt Gribble, C++ program
FORMULA
A224850(n,k) + A224861(n,k) + T(n,k) = A227690(n,k).
1*A224850(n,k) + 2*A224861(n,k) + 4*T(n,k) = A219924(n,k).
EXAMPLE
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:
._________. ._________. ._________. ._________. ._________.
| |_|_|_| |_| |_|_| | | |_| | |_|_|_| | | |
|_ _|_|_|_| |_|_ _|_|_| |_ _|_ _|_| |___| |_| |___| |
|_|_|_|_|_| |_|_|_|_|_| |_|_|_|_|_| |_|_|___|_| |_|_|_____|
CROSSREFS
KEYWORD
nonn,tabl,more
AUTHOR
STATUS
approved