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%I #30 Sep 06 2021 04:50:57
%S 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,
%T 11,106,44
%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 1 element; triangle T(n,k), k >= 1, 0 <= n < k, read by columns.
%C It appears that sequence T(2,k) consists of 2 interspersed Fibonacci sequences.
%C The diagonal T(n,n) is A006081. - _M. F. Hasler_, Jul 25 2013
%H Christopher Hunt Gribble, <a href="/A224850/a224850.cpp.txt">C++ program</a>
%F T(n,k) + A224861(n,k) + A224867(n,k) = A227690(n,k).
%F 1*T(n,k) + 2*A224861(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 1 1 1 1 1 1 1 1 ...
%e 1 1 1 1 1 1 1 1 ...
%e 2 1 3 2 5 3 8 ...
%e 3 1 2 2 3 3 ...
%e 4 3 12 5 25 ...
%e 5 6 7 11 ...
%e 6 17 106 ...
%e 7 44 ...
%e ...
%e 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:
%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 The tilings are:
%e ._________. ._________.
%e |_|_|_|_|_| |_| |_|
%e |_|_|_|_|_| |_| |_|
%e |_|_|_|_|_| |_|_____|_|
%Y Cf. A219924, A224697, A227690.
%K nonn,tabl,more
%O 1,9
%A _Christopher Hunt Gribble_, Jul 22 2013
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