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 A361216 #17 Mar 12 2023 10:45:05 %S A361216 1,1,4,2,11,56,3,29,370,5752,4,94,2666,82310,2519124,6,263,19126, %T A361216 1232770,88117873,6126859968,12,968,134902,19119198,2835424200 %N A361216 Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X k rectangle. %C A361216 Tilings that are rotations or reflections of each other are considered distinct. %C A361216 Pieces can have any combination of integer side lengths, but for the optimal sets computed so far (up to (n,k) = (7,5)), all pieces have one side of length 1. %F A361216 T(n,1) = A102462(n). %e A361216 Triangle begins: %e A361216 n\k| 1 2 3 4 5 6 7 8 %e A361216 ---+-------------------------------------------------------- %e A361216 1 | 1 %e A361216 2 | 1 4 %e A361216 3 | 2 11 56 %e A361216 4 | 3 29 370 5752 %e A361216 5 | 4 94 2666 82310 2519124 %e A361216 6 | 6 263 19126 1232770 88117873 6126859968 %e A361216 7 | 12 968 134902 19119198 2835424200 ? ? %e A361216 8 | 20 3416 1026667 307914196 109979838540 ? ? ? %e A361216 A 3 X 3 square can be tiled by three 1 X 2 pieces and three 1 X 1 pieces in the following ways: %e A361216 +---+---+---+ +---+---+---+ +---+---+---+ %e A361216 | | | | | | | | | | | | %e A361216 +---+---+---+ + +---+---+ +---+ +---+ %e A361216 | | | | | | | | | | | %e A361216 +---+---+ + +---+---+ + +---+---+ + %e A361216 | | | | | | | | | %e A361216 +---+---+---+ +---+---+---+ +---+---+---+ %e A361216 . %e A361216 +---+---+---+ +---+---+---+ +---+---+---+ %e A361216 | | | | | | | | | | %e A361216 +---+---+---+ +---+---+ + +---+---+---+ %e A361216 | | | | | | | | | | %e A361216 +---+---+ + +---+---+---+ +---+---+---+ %e A361216 | | | | | | | | | %e A361216 +---+---+---+ +---+---+---+ +---+---+---+ %e A361216 . %e A361216 +---+---+---+ +---+---+---+ %e A361216 | | | | | | %e A361216 +---+---+---+ +---+---+---+ %e A361216 | | | | | | %e A361216 +---+---+---+ +---+---+---+ %e A361216 | | | | | | %e A361216 +---+---+---+ +---+---+---+ %e A361216 The first six of these have no symmetries, so they account for 8 tilings each. The last two has a mirror symmetry, so they account for 4 tilings each. In total there are 6*8+2*4 = 56 tilings. This is the maximum for a 3 X 3 square, so T(3,3) = 56. %e A361216 The following table shows the sets of pieces that give the maximum number of tilings up to (n,k) = (7,5). The solutions are unique except for (n,k) = (2,1) and (n,k) = (6,1). %e A361216 \ Number of pieces of size %e A361216 (n,k)\ 1 X 1 | 1 X 2 | 1 X 3 | 1 X 4 %e A361216 ------+-------+-------+-------+------ %e A361216 (1,1) | 1 | 0 | 0 | 0 %e A361216 (2,1) | 2 | 0 | 0 | 0 %e A361216 (2,1) | 0 | 1 | 0 | 0 %e A361216 (2,2) | 2 | 1 | 0 | 0 %e A361216 (3,1) | 1 | 1 | 0 | 0 %e A361216 (3,2) | 2 | 2 | 0 | 0 %e A361216 (3,3) | 3 | 3 | 0 | 0 %e A361216 (4,1) | 2 | 1 | 0 | 0 %e A361216 (4,2) | 4 | 2 | 0 | 0 %e A361216 (4,3) | 3 | 3 | 1 | 0 %e A361216 (4,4) | 5 | 4 | 1 | 0 %e A361216 (5,1) | 3 | 1 | 0 | 0 %e A361216 (5,2) | 4 | 3 | 0 | 0 %e A361216 (5,3) | 4 | 4 | 1 | 0 %e A361216 (5,4) | 7 | 5 | 1 | 0 %e A361216 (5,5) | 7 | 6 | 2 | 0 %e A361216 (6,1) | 2 | 2 | 0 | 0 %e A361216 (6,1) | 1 | 1 | 1 | 0 %e A361216 (6,2) | 4 | 4 | 0 | 0 %e A361216 (6,3) | 7 | 4 | 1 | 0 %e A361216 (6,4) | 8 | 5 | 2 | 0 %e A361216 (6,5) | 10 | 7 | 2 | 0 %e A361216 (6,6) | 11 | 8 | 3 | 0 %e A361216 (7,1) | 2 | 1 | 1 | 0 %e A361216 (7,2) | 5 | 3 | 1 | 0 %e A361216 (7,3) | 8 | 5 | 1 | 0 %e A361216 (7,4) | 10 | 6 | 2 | 0 %e A361216 (7,5) | 11 | 7 | 2 | 1 %Y A361216 Main diagonal: A361217. %Y A361216 Columns: A102462 (k = 1), A361218 (k = 2), A361219 (k = 3), A361220 (k = 4). %Y A361216 Cf. A360629, A361221. %K A361216 nonn,tabl,more %O A361216 1,3 %A A361216 _Pontus von Brömssen_, Mar 05 2023