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%I #12 Feb 16 2023 11:31:59
%S 1,1,21,269,4489,82981,2995185,118897973
%N Number of ways to tile an n X n square using rectangles with distinct dimensions.
%C All possible tilings are counted, including those identical by symmetry. Note that distinct dimensions means that, for example, a 1 x 3 rectangle can only be used once, regardless of if it lies horizontally or vertically.
%e a(1) = 1 as the only way to tile a 1 x 1 square is with a square with dimensions 1 x 1.
%e a(2) = 1 as the only way to tile a 2 x 2 square is with a square with dimensions 2 x 2.
%e a(3) = 21. The possible tilings, excluding those equivalent by symmetry, are:
%e .
%e +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+
%e | | | | | | | | | |
%e + + +---+---+---+ +---+---+ + +---+---+---+
%e | | | | | | | | |
%e + + + + + + + + +
%e | | | | | | | | |
%e +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+
%e .
%e The first tiling can occur in 1 way, the second in 8 different ways, the third in 8 different ways and the fourth in 4 different ways, giving 21 ways in total.
%Y Cf. A360498 (oblongs), A182275 (not necessarily distinct dimensions), A004003, A099390, A065072, A233320, A230031.
%K nonn,more
%O 1,3
%A _Scott R. Shannon_, Feb 09 2023