%I #4 Feb 18 2023 08:08:43
%S 0,0,4,36,1056,31052,1473944,87469884
%N Number of ways to tile an n X n square using oblongs with distinct height x width dimensions.
%C 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 a(1) = 0 as no distinct oblongs can tile a square with dimensions 1 x 1.
%e a(2) = 0 as no distinct oblongs can tile a square with dimensions 2 x 2.
%e a(3) = 4. There is one tiling, excluding those equivalent by symmetry:
%e .
%e +---+---+---+
%e | |
%e +---+---+---+
%e | |
%e + +
%e | |
%e +---+---+---+
%e .
%e This tiling can occur in 4 different ways, giving 4 ways in total.
%e a(4) = 36. The possible tilings, excluding those equivalent by symmetry, are:
%e .
%e +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
%e | | | | | | | | | | |
%e + + + +---+---+---+---+ + +---+---+---+ + +---+---+---+
%e | | | | | | | | | | | |
%e +---+---+---+---+ + + + + + + + + +
%e | | | | | | | | | | |
%e + + + + +---+---+---+---+ +---+---+ +
%e | | | | | | | | |
%e +---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
%e .
%e 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 Cf. A360256 (rectangles), A360499, A360498, A182275, A004003, A099390, A065072.
%K nonn,more
%O 1,3
%A _Scott R. Shannon_, Feb 18 2023