This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A224867 #26 Sep 06 2021 04:55:55 %S A224867 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 A224867 14508,0,0,0,58,676,7716,81119,856559 %N 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. %H A224867 Christopher Hunt Gribble, <a href="/A224867/a224867.cpp.txt">C++ program</a> %F A224867 A224850(n,k) + A224861(n,k) + T(n,k) = A227690(n,k). %F A224867 1*A224850(n,k) + 2*A224861(n,k) + 4*T(n,k) = A219924(n,k). %e A224867 The triangle is: %e A224867 n\k 1 2 3 4 5 6 7 8 ... %e A224867 . %e A224867 0 0 0 0 0 0 0 0 0 ... %e A224867 1 0 0 0 0 0 0 0 ... %e A224867 2 0 0 0 0 0 0 ... %e A224867 3 1 5 10 27 58 ... %e A224867 4 21 65 222 676 ... %e A224867 5 440 1901 7716 ... %e A224867 6 14508 81119 ... %e A224867 7 856559 ... %e A224867 ... %e A224867 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 A224867 . the identity operation %e A224867 . rotation by 180 degrees %e A224867 . reflection about a horizontal axis through the center %e A224867 . reflection about a vertical axis through the center %e A224867 An example of a tiling in each set is: %e A224867 ._________. ._________. ._________. ._________. ._________. %e A224867 | |_|_|_| |_| |_|_| | | |_| | |_|_|_| | | | %e A224867 |_ _|_|_|_| |_|_ _|_|_| |_ _|_ _|_| |___| |_| |___| | %e A224867 |_|_|_|_|_| |_|_|_|_|_| |_|_|_|_|_| |_|_|___|_| |_|_|_____| %Y A224867 Cf. A219924, A224697, A227690. %K A224867 nonn,tabl,more %O A224867 1,14 %A A224867 _Christopher Hunt Gribble_, Jul 22 2013