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 A360629 #23 May 11 2025 14:20:56 %S A360629 1,2,4,3,10,21,5,22,73,192,7,44,190,703,2035,11,91,510,2287,8581, %T A360629 27407,15,172,1196,6738,30209,118461,399618,22,326,2895,19160,102092, %U A360629 462114 %N A360629 Triangle read by rows: T(n,k) is the number of sets of integer-sided rectangular pieces that can tile an n X k rectangle, 1 <= k <= n. %C A360629 Pieces are free to rotate by 90 degrees, i.e., an r X s piece and an s X r piece are equivalent. See A360451 for the case when the pieces are fixed. %e A360629 Triangle begins: %e A360629 n\k| 1 2 3 4 5 6 7 %e A360629 ---+-------------------------------------- %e A360629 1 | 1 %e A360629 2 | 2 4 %e A360629 3 | 3 10 21 %e A360629 4 | 5 22 73 192 %e A360629 5 | 7 44 190 703 2035 %e A360629 6 | 11 91 510 2287 8581 27407 %e A360629 7 | 15 172 1196 6738 30209 118461 399618 %e A360629 ... %e A360629 T(2,2) = 4, because a 2 X 2 rectangle can be tiled by: one 2 X 2 piece; two 1 X 2 pieces; one 1 X 2 piece and two 1 X 1 pieces; four 1 X 1 pieces. %e A360629 The T(3,2) = 10 sets of pieces that can tile a 3 X 2 rectangle are shown in the table below. (Each column on the right gives a set of pieces.) %e A360629 length X width | number of pieces %e A360629 ---------------+-------------------- %e A360629 2 X 3 | 1 0 0 0 0 0 0 0 0 0 %e A360629 2 X 2 | 0 1 1 0 0 0 0 0 0 0 %e A360629 1 X 3 | 0 0 0 2 1 1 0 0 0 0 %e A360629 1 X 2 | 0 1 0 0 1 0 3 2 1 0 %e A360629 1 X 1 | 0 0 2 0 1 3 0 2 4 6 %Y A360629 Cf. A000041 (column k=1), A116694, A224697 (square pieces), A360451 (fixed pieces), A360630 (main diagonal), A360631 (column k=2), A360632 (column k=3). %K A360629 nonn,tabl,more %O A360629 1,2 %A A360629 _Pontus von Brömssen_, Feb 14 2023 %E A360629 T(7,7) and T(8,k) for k = 1..6 added by _Robin Visser_, May 09 2025