%I #11 Dec 30 2023 17:03:27
%S 0,0,4,12,256,3620,87216,2444084,87181220
%N Number of ways to tile an n X n square using oblongs 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 oblong can only be used once, regardless of if it lies horizontally or vertically.
%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) = 12. 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 way and the second in 4 different ways, giving 12 ways in total.
%Y Cf. A360499 (rectangles), A004003, A099390, A065072, A233320, A230031.
%K nonn,more
%O 1,3
%A _Scott R. Shannon_, Feb 09 2023