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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.
4
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
1,10
LINKS
Christopher Hunt Gribble, C++ program
FORMULA
A224850(n,k) + T(n,k) + A224867(n,k) = A227690(n,k).
1*A224850(n,k) + 2*T(n,k) + 4*A224867(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 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:
._________. ._________. ._________.
| |_| | | |_|_|_| | |_|_|
|_ _|_|_ _| |___|_| | | |_|_|
|_|_|_|_|_| |_|_|_|___| |_____|_|_|
CROSSREFS
KEYWORD
nonn,tabl,more
AUTHOR
STATUS
approved