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 A361224 #6 Mar 11 2023 08:38:21 %S A361224 1,1,5,12,31,86,242,854,2888,10478,34264,120347 %N A361224 Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X 2 rectangle, up to rotations and reflections. %e A361224 A 4 X 2 rectangle can be tiled by two 1 X 2 pieces and four 1 X 1 pieces in the following 12 ways: %e A361224 +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ %e A361224 | | | | | | | | | | | | | | | | | %e A361224 +---+---+ +---+---+ +---+---+ + +---+ +---+---+ +---+---+ %e A361224 | | | | | | | | | | | | | | | | | %e A361224 +---+---+ + +---+ +---+---+ +---+---+ +---+---+ +---+---+ %e A361224 | | | | | | | | | | | | | | | | | %e A361224 +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ + + + %e A361224 | | | | | | | | | | | | | %e A361224 +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ %e A361224 . %e A361224 +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ %e A361224 | | | | | | | | | | | | | | | | | | %e A361224 +---+---+ +---+---+ + +---+ +---+ + +---+---+ +---+---+ %e A361224 | | | | | | | | | | | | | | | | %e A361224 +---+ + +---+---+ +---+---+ +---+---+ +---+---+ + + + %e A361224 | | | | | | | | | | | | | | | | | %e A361224 + +---+ + +---+ + +---+ + +---+ +---+---+ +---+---+ %e A361224 | | | | | | | | | | | | | | | | | | %e A361224 +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ %e A361224 This is the maximum for a 4 X 2 rectangle, so a(4) = 12. %e A361224 The following table shows the sets of pieces that give the maximum number of tilings for n <= 12. The solutions are unique except for n <= 2. %e A361224 \ Number of pieces of size %e A361224 n \ 1 X 1 | 1 X 2 | 1 X 3 | 2 X 2 %e A361224 ----+-------+-------+-------+------ %e A361224 1 | 2 | 0 | 0 | 0 %e A361224 1 | 0 | 1 | 0 | 0 %e A361224 2 | 4 | 0 | 0 | 0 %e A361224 2 | 2 | 1 | 0 | 0 %e A361224 2 | 0 | 2 | 0 | 0 %e A361224 2 | 0 | 0 | 0 | 1 %e A361224 3 | 2 | 2 | 0 | 0 %e A361224 4 | 4 | 2 | 0 | 0 %e A361224 5 | 4 | 3 | 0 | 0 %e A361224 6 | 4 | 4 | 0 | 0 %e A361224 7 | 5 | 3 | 1 | 0 %e A361224 8 | 5 | 4 | 1 | 0 %e A361224 9 | 7 | 4 | 1 | 0 %e A361224 10 | 7 | 5 | 1 | 0 %e A361224 11 | 7 | 6 | 1 | 0 %e A361224 12 | 9 | 6 | 1 | 0 %e A361224 It seems that all optimal solutions for A361218 are also optimal here, but for n = 2 there are other optimal solutions. %Y A361224 Second column of A361221. %Y A361224 Cf. A361218, A360631. %K A361224 nonn,more %O A361224 1,3 %A A361224 _Pontus von Brömssen_, Mar 05 2023