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 A224850 #30 Sep 06 2021 04:50:57 %S A224850 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 A224850 11,106,44 %N A224850 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 A224850 It appears that sequence T(2,k) consists of 2 interspersed Fibonacci sequences. %C A224850 The diagonal T(n,n) is A006081. - _M. F. Hasler_, Jul 25 2013 %H A224850 Christopher Hunt Gribble, <a href="/A224850/a224850.cpp.txt">C++ program</a> %F A224850 T(n,k) + A224861(n,k) + A224867(n,k) = A227690(n,k). %F A224850 1*T(n,k) + 2*A224861(n,k) + 4*A224867(n,k) = A219924(n,k). %e A224850 The triangle is: %e A224850 n\k 1 2 3 4 5 6 7 8 ... %e A224850 . %e A224850 0 1 1 1 1 1 1 1 1 ... %e A224850 1 1 1 1 1 1 1 1 ... %e A224850 2 1 3 2 5 3 8 ... %e A224850 3 1 2 2 3 3 ... %e A224850 4 3 12 5 25 ... %e A224850 5 6 7 11 ... %e A224850 6 17 106 ... %e A224850 7 44 ... %e A224850 ... %e A224850 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 A224850 . the identity operation %e A224850 . rotation by 180 degrees %e A224850 . reflection about a horizontal axis through the center %e A224850 . reflection about a vertical axis through the center %e A224850 The tilings are: %e A224850 ._________. ._________. %e A224850 |_|_|_|_|_| |_| |_| %e A224850 |_|_|_|_|_| |_| |_| %e A224850 |_|_|_|_|_| |_|_____|_| %Y A224850 Cf. A219924, A224697, A227690. %K A224850 nonn,tabl,more %O A224850 1,9 %A A224850 _Christopher Hunt Gribble_, Jul 22 2013