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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,14

COMMENTS

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 ...

LINKS

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

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

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

Cf. A219924, A224697, A227690.

Sequence in context: A213148 A202860 A156148 * A156824 A053002 A053003

Adjacent sequences:  A224864 A224865 A224866 * A224868 A224869 A224870

KEYWORD

nonn,tabl

AUTHOR

Christopher Hunt Gribble, Jul 22 2013

STATUS

approved

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Last modified July 21 05:11 EDT 2019. Contains 325189 sequences. (Running on oeis4.)