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

1,10

Links

Table of n, a(n) for n=1..36.

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

Cf. A219924, A224697, A227690.

Sequence in context: A325492 A266270 A091467 * A050443 A306770 A308278

Adjacent sequences: A224858 A224859 A224860 * A224862 A224863 A224864

Keyword

nonn,tabl,more

Author

Christopher Hunt Gribble, Jul 22 2013

Status

approved