A224850 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 1 element; triangle T(n,k), k >= 1, 0 <= n < k, read by columns.

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 3, 1, 1, 5, 2, 12, 6, 1, 1, 3, 3, 5, 7, 17, 1, 1, 8, 3, 25, 11, 106, 44

Offset

1,9

Comments

It appears that sequence T(2,k) consists of 2 interspersed Fibonacci sequences.

The diagonal T(n,n) is A006081. - M. F. Hasler, Jul 25 2013

Links

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

Christopher Hunt Gribble, C++ program

Formula

T(n,k) + A224861(n,k) + A224867(n,k) = A227690(n,k).

1*T(n,k) + 2*A224861(n,k) + 4*A224867(n,k) = A219924(n,k).

Example

The triangle is:
n\k  1   2   3   4   5   6   7   8 ...
.
0    1   1   1   1   1   1   1   1 ...
1        1   1   1   1   1   1   1 ...
2            1   3   2   5   3   8 ...
3                1   2   2   3   3 ...
4                    3  12   5  25 ...
5                        6   7  11 ...
6                           17 106 ...
7                               44 ...
...
T(3,5) = 2 because there are 2 different tilings of the 3 X 5 rectangle by integer-sided squares, where any sequence of group D2 operations will only transform each tiling into itself.  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
The tilings are:
._________.    ._________.
|_|_|_|_|_|    |_|     |_|
|_|_|_|_|_|    |_|     |_|
|_|_|_|_|_|    |_|_____|_|

Crossrefs

Cf. A219924, A224697, A227690.

Sequence in context: A178252 A338199 A325498 * A269976 A132409 A030337

Adjacent sequences: A224847 A224848 A224849 * A224851 A224852 A224853

Keyword

nonn,tabl,more

Author

Christopher Hunt Gribble, Jul 22 2013

Status

approved