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 A224861 #26 Sep 06 2021 04:29:59 %S A224861 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 A224861 13,21,160,311,2384,4556 %N A224861 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 A224861 Christopher Hunt Gribble, <a href="/A224861/a224861.cpp.txt">C++ program</a> %F A224861 A224850(n,k) + T(n,k) + A224867(n,k) = A227690(n,k). %F A224861 1*A224850(n,k) + 2*T(n,k) + 4*A224867(n,k) = A219924(n,k). %e A224861 The triangle is: %e A224861 n\k 1 2 3 4 5 6 7 8 ... %e A224861 . %e A224861 0 0 0 0 0 0 0 0 0 ... %e A224861 1 0 0 0 0 0 0 0 ... %e A224861 2 1 1 3 4 9 13 ... %e A224861 3 4 3 9 9 21 ... %e A224861 4 15 38 68 160 ... %e A224861 5 75 77 311 ... %e A224861 6 604 2384 ... %e A224861 7 4556 ... %e A224861 ... %e A224861 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 A224861 . the identity operation %e A224861 . rotation by 180 degrees %e A224861 . reflection about a horizontal axis through the center %e A224861 . reflection about a vertical axis through the center %e A224861 An example of a tiling in each set is: %e A224861 ._________. ._________. ._________. %e A224861 | |_| | | |_|_|_| | |_|_| %e A224861 |_ _|_|_ _| |___|_| | | |_|_| %e A224861 |_|_|_|_|_| |_|_|_|___| |_____|_|_| %Y A224861 Cf. A219924, A224697, A227690. %K A224861 nonn,tabl,more %O A224861 1,10 %A A224861 _Christopher Hunt Gribble_, Jul 22 2013