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 A361221 #7 Mar 11 2023 08:38:06 %S A361221 1,1,1,1,5,8,2,12,95,719,2,31,682,20600,315107 %N A361221 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, up to rotations and reflections. %e A361221 Triangle begins: %e A361221 n\k| 1 2 3 4 5 %e A361221 ---+------------------------ %e A361221 1 | 1 %e A361221 2 | 1 1 %e A361221 3 | 1 5 8 %e A361221 4 | 2 12 95 719 %e A361221 5 | 2 31 682 20600 315107 %e A361221 A 3 X 3 square can be tiled by one 1 X 3 piece, two 1 X 2 pieces and two 1 X 1 pieces in the following 8 ways: %e A361221 +---+---+---+ +---+---+---+ +---+---+---+ %e A361221 | | | | | | | | | | %e A361221 +---+---+ + +---+---+---+ +---+---+---+ %e A361221 | | | | | | | | | %e A361221 +---+---+---+ +---+---+---+ +---+---+---+ %e A361221 | | | | | | %e A361221 +---+---+---+ +---+---+---+ +---+---+---+ %e A361221 . %e A361221 +---+---+---+ +---+---+---+ +---+---+---+ %e A361221 | | | | | | | | | | | %e A361221 + + +---+ + +---+ + + +---+---+ %e A361221 | | | | | | | | | | | | %e A361221 +---+---+---+ +---+---+---+ +---+---+---+ %e A361221 | | | | | | %e A361221 +---+---+---+ +---+---+---+ +---+---+---+ %e A361221 . %e A361221 +---+---+---+ +---+---+---+ %e A361221 | | | | | | %e A361221 +---+---+---+ +---+---+---+ %e A361221 | | | | %e A361221 +---+---+---+ +---+---+---+ %e A361221 | | | | | | %e A361221 +---+---+---+ +---+---+---+ %e A361221 This is the maximum for a 3 X 3 square, so T(3,3) = 8. There is one other set of pieces that also can tile the 3 X 3 square in 8 ways: three 1 X 2 pieces and three 1 X 1 pieces (see illustration in A361216). %e A361221 The following table shows all sets of pieces that give the maximum number of tilings for 1 <= k <= n <= 5: %e A361221 \ Number of pieces of size %e A361221 (n,k)\ 1 X 1 | 1 X 2 | 1 X 3 | 2 X 2 %e A361221 ------+-------+-------+-------+------ %e A361221 (1,1) | 1 | 0 | 0 | 0 %e A361221 (2,1) | 2 | 0 | 0 | 0 %e A361221 (2,1) | 0 | 1 | 0 | 0 %e A361221 (2,2) | 4 | 0 | 0 | 0 %e A361221 (2,2) | 2 | 1 | 0 | 0 %e A361221 (2,2) | 0 | 2 | 0 | 0 %e A361221 (2,2) | 0 | 0 | 0 | 1 %e A361221 (3,1) | 3 | 0 | 0 | 0 %e A361221 (3,1) | 1 | 1 | 0 | 0 %e A361221 (3,1) | 0 | 0 | 1 | 0 %e A361221 (3,2) | 2 | 2 | 0 | 0 %e A361221 (3,3) | 3 | 3 | 0 | 0 %e A361221 (3,3) | 2 | 2 | 1 | 0 %e A361221 (4,1) | 2 | 1 | 0 | 0 %e A361221 (4,2) | 4 | 2 | 0 | 0 %e A361221 (4,3) | 3 | 3 | 1 | 0 %e A361221 (4,4) | 5 | 4 | 1 | 0 %e A361221 (5,1) | 3 | 1 | 0 | 0 %e A361221 (5,1) | 2 | 0 | 1 | 0 %e A361221 (5,1) | 1 | 2 | 0 | 0 %e A361221 (5,2) | 4 | 3 | 0 | 0 %e A361221 (5,3) | 4 | 4 | 1 | 0 %e A361221 (5,4) | 7 | 5 | 1 | 0 %e A361221 (5,5) | 7 | 6 | 2 | 0 %e A361221 It seems that all optimal solutions for A361216 are also optimal here, but occasionally there are other optimal solutions, e.g. for n = k = 3. %Y A361221 Main diagonal: A361222. %Y A361221 Columns: A361223 (k = 1), A361224 (k = 2), A361225 (k = 3). %Y A361221 Cf. A360629, A361216. %K A361221 nonn,tabl,more %O A361221 1,5 %A A361221 _Pontus von Brömssen_, Mar 05 2023