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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

%I #26 Sep 06 2021 04:55:55

%S 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,

%T 14508,0,0,0,58,676,7716,81119,856559

%N 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.

%H Christopher Hunt Gribble, <a href="/A224867/a224867.cpp.txt">C++ program</a>

%F A224850(n,k) + A224861(n,k) + T(n,k) = A227690(n,k).

%F 1*A224850(n,k) + 2*A224861(n,k) + 4*T(n,k) = A219924(n,k).

%e The triangle is:

%e n\k 1 2 3 4 5 6 7 8 ...

%e .

%e 0 0 0 0 0 0 0 0 0 ...

%e 1 0 0 0 0 0 0 0 ...

%e 2 0 0 0 0 0 0 ...

%e 3 1 5 10 27 58 ...

%e 4 21 65 222 676 ...

%e 5 440 1901 7716 ...

%e 6 14508 81119 ...

%e 7 856559 ...

%e ...

%e 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:

%e . the identity operation

%e . rotation by 180 degrees

%e . reflection about a horizontal axis through the center

%e . reflection about a vertical axis through the center

%e An example of a tiling in each set is:

%e ._________. ._________. ._________. ._________. ._________.

%e | |_|_|_| |_| |_|_| | | |_| | |_|_|_| | | |

%e |_ _|_|_|_| |_|_ _|_|_| |_ _|_ _|_| |___| |_| |___| |

%e |_|_|_|_|_| |_|_|_|_|_| |_|_|_|_|_| |_|_|___|_| |_|_|_____|

%Y Cf. A219924, A224697, A227690.

%K nonn,tabl,more

%O 1,14

%A _Christopher Hunt Gribble_, Jul 22 2013