cp's OEIS Frontend

A360804 Number of ways to tile an n X n square using rectangles with distinct areas.

%I A360804 #10 Feb 25 2023 08:33:02
%S A360804 1,1,21,253,2401,36237,815929,18713197
%N A360804 Number of ways to tile an n X n square using rectangles with distinct areas.
%C A360804 All possible tilings are counted, including those identical by symmetry. Note that distinct areas means that, for example, only one of the two rectangles with area 4, a 2 X 2 or 1 X 4 rectangle, can be used in any tiling.
%e A360804 a(1) = 1 as the only way to tile a 1 X 1 square is with a square with dimensions 1 X 1.
%e A360804 a(2) = 1 as the only way to tile a 2 X 2 square is with a square with dimensions 2 X 2.
%e A360804 a(3) = 21. The possible tilings are the same as those given in the examples of A360499(3).
%e A360804 a(4) = 253. And example tiling of the 4 X 4 square is:
%e A360804 .
%e A360804   +---+---+---+---+
%e A360804   |   |       |   |
%e A360804   +---+---+---+   +
%e A360804   |           |   |
%e A360804   +           +   +
%e A360804   |           |   |
%e A360804   +---+---+---+---+
%e A360804   |               |
%e A360804   +---+---+---+---+
%e A360804 .
%e A360804 which contains rectangles with areas 1, 2, 3, 4, 6. The one tiling, excluding symmetrically equivalent arrangements, that is excluded here but allowed in A360499 is:
%e A360804 .
%e A360804   +---+---+---+---+
%e A360804   |       |       |
%e A360804   +       +       +
%e A360804   |       |       |
%e A360804   +---+---+       +
%e A360804   |       |       |
%e A360804   +---+---+---+---+
%e A360804   |               |
%e A360804   +---+---+---+---+
%e A360804 .
%e A360804 as this contains two rectangles with area 4. This can occur in 16 different ways so a(4) = A360499(4) - 16 = 269 - 16 = 253.
%Y A360804 Cf. A360499, A360498, A360725, A360256, A360773, A182275, A004003.
%K A360804 nonn,more
%O A360804 1,3
%A A360804 _Scott R. Shannon_, Feb 21 2023