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%I #26 Sep 06 2021 04:29:59
%S 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,
%T 13,21,160,311,2384,4556
%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 2 elements; triangle T(n,k), k >= 1, 0 <= n < k, read by columns.
%H Christopher Hunt Gribble, <a href="/A224861/a224861.cpp.txt">C++ program</a>
%F A224850(n,k) + T(n,k) + A224867(n,k) = A227690(n,k).
%F 1*A224850(n,k) + 2*T(n,k) + 4*A224867(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 1 1 3 4 9 13 ...
%e 3 4 3 9 9 21 ...
%e 4 15 38 68 160 ...
%e 5 75 77 311 ...
%e 6 604 2384 ...
%e 7 4556 ...
%e ...
%e 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:
%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 1,10
%A _Christopher Hunt Gribble_, Jul 22 2013