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A360725 Number of ways to tile an n X n square using oblongs with distinct height x width dimensions.

%I A360725 #4 Feb 18 2023 08:08:43
%S A360725 0,0,4,36,1056,31052,1473944,87469884
%N A360725 Number of ways to tile an n X n square using oblongs with distinct height x width dimensions.
%C A360725 All possible tilings are counted, including those identical by symmetry. Note that distinct height x width dimensions means that, for example, a 1 x 3 oblong can be used twice, once in a horizonal (1 x 3) and once in a vertical (3 x 1) direction.
%e A360725 a(1) = 0 as no distinct oblongs can tile a square with dimensions 1 x 1.
%e A360725 a(2) = 0 as no distinct oblongs can tile a square with dimensions 2 x 2.
%e A360725 a(3) = 4. There is one tiling, excluding those equivalent by symmetry:
%e A360725 .
%e A360725   +---+---+---+
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%e A360725 .
%e A360725 This tiling can occur in 4 different ways, giving 4 ways in total.
%e A360725 a(4) = 36. The possible tilings, excluding those equivalent by symmetry, are:
%e A360725 .
%e A360725   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
%e A360725   |   |           |   |               |   |   |           |   |   |           |
%e A360725   +   +           +   +---+---+---+---+   +   +---+---+---+   +   +---+---+---+
%e A360725   |   |           |   |               |   |   |           |   |   |   |       |
%e A360725   +---+---+---+---+   +               +   +   +           +   +   +   +       +
%e A360725   |               |   |               |   |   |           |   |   |   |       |
%e A360725   +               +   +               +   +---+---+---+---+   +---+---+       +
%e A360725   |               |   |               |   |               |   |       |       |
%e A360725   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
%e A360725 .
%e A360725 The first tiling can occur in 8 different ways, the second in 4 different ways, the third in 16 different ways and the fourth in 8 different ways. This gives 36 ways in total.
%Y A360725 Cf. A360256 (rectangles), A360499, A360498, A182275, A004003, A099390, A065072.
%K A360725 nonn,more
%O A360725 1,3
%A A360725 _Scott R. Shannon_, Feb 18 2023