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 A362142 #12 Apr 11 2023 11:54:16 %S A362142 1,1,1,1,1,1,1,1,2,4,1,1,3,6,16,1,1,4,12,37,140,1,1,6,24,105,454,1987, %T A362142 1,1,10,40,250,1566,9856,62266,1,1,15,80,726,5670,47394,406168, %U A362142 3899340,1,1,21,160,1824,18738,223696,2916492,38322758,508317004 %N A362142 Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided squares can tile an n X k rectangle, 0 <= k <= n. %H A362142 Pontus von Brömssen, <a href="/A362142/b362142.txt">Table of n, a(n) for n = 0..90</a> (rows 0..12) %e A362142 Triangle begins: %e A362142 n\k| 0 1 2 3 4 5 6 7 8 %e A362142 ---+----------------------------------------- %e A362142 0 | 1 %e A362142 1 | 1 1 %e A362142 2 | 1 1 1 %e A362142 3 | 1 1 2 4 %e A362142 4 | 1 1 3 6 16 %e A362142 5 | 1 1 4 12 37 140 %e A362142 6 | 1 1 6 24 105 454 1987 %e A362142 7 | 1 1 10 40 250 1566 9856 62266 %e A362142 8 | 1 1 15 80 726 5670 47394 406168 3899340 %e A362142 A 5 X 4 rectangle can be tiled by 12 unit squares and 2 squares of side 2 in the following ways: %e A362142 +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ %e A362142 | | | | | | | | | | | | | | | | | | | | %e A362142 +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ %e A362142 | | | | | | | | | | | | | | | | | %e A362142 +---+---+---+---+ + +---+---+ +---+ +---+ +---+---+ + %e A362142 | | | | | | | | | | | | | | | | %e A362142 +---+---+ + +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ %e A362142 | | | | | | | | | | | | | | | %e A362142 + +---+---+ + +---+---+ + +---+---+ + +---+---+ %e A362142 | | | | | | | | | | | | | | | | %e A362142 +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ %e A362142 . %e A362142 +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ %e A362142 | | | | | | | | | | | | | | | | | | %e A362142 +---+ +---+ +---+---+---+---+ +---+---+---+---+ + +---+---+ %e A362142 | | | | | | | | | | | | | | | | | %e A362142 +---+---+---+---+ + +---+---+ +---+---+---+---+ +---+---+---+---+ %e A362142 | | | | | | | | | | | | | | | | | | | %e A362142 +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ %e A362142 | | | | | | | | | | | | | | | %e A362142 + +---+---+ +---+ +---+ + + + + +---+---+ %e A362142 | | | | | | | | | | | | | | | %e A362142 +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ %e A362142 . %e A362142 +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ %e A362142 | | | | | | | | | | | | | | | | | | | %e A362142 +---+---+ + +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ %e A362142 | | | | | | | | | | | | | | | | | %e A362142 +---+---+---+---+ +---+ +---+ +---+---+---+---+ +---+---+ + %e A362142 | | | | | | | | | | | | | | | %e A362142 +---+---+---+---+ +---+---+---+---+ + + + + +---+---+ %e A362142 | | | | | | | | | | | | | | | %e A362142 + +---+---+ +---+ +---+ +---+---+---+---+ +---+---+---+---+ %e A362142 | | | | | | | | | | | | | | | | | | %e A362142 +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ %e A362142 . %e A362142 +---+---+---+---+ %e A362142 | | | | %e A362142 +---+ +---+ %e A362142 | | | | %e A362142 +---+---+---+---+ %e A362142 | | | | | %e A362142 +---+---+---+---+ %e A362142 | | | | %e A362142 +---+ +---+ %e A362142 | | | | %e A362142 +---+---+---+---+ %e A362142 The first six of these have no symmetries, so they account for 4 tilings each. The next six have either a mirror symmetry or a rotational symmetry and account for 2 tilings each. The last has full symmetry and accounts for 1 tiling. In total there are 6*4+6*2+1 = 37 tilings. This is the maximum for a 5 X 4 rectangle, so T(5,4) = 37. %Y A362142 Main diagonal: A362143. %Y A362142 Columns: A000012 (k = 0,1), A073028 (k = 2), A362144 (k = 3), A362145 (k = 4), A362146 (k = 5). %Y A362142 Cf. A219924, A224697, A361216 (rectangular pieces). %K A362142 nonn,tabl %O A362142 0,9 %A A362142 _Pontus von Brömssen_, Apr 10 2023