A360804 - OEIS

%I #10 Feb 25 2023 08:33:02

%S 1,1,21,253,2401,36237,815929,18713197

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

%C 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 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 are the same as those given in the examples of A360499(3).

%e a(4) = 253. And example tiling of the 4 X 4 square is:

%e .

%e   +---+---+---+---+

%e   |   |       |   |

%e   +---+---+---+   +

%e   |           |   |

%e   +           +   +

%e   |           |   |

%e   +---+---+---+---+

%e   |               |

%e   +---+---+---+---+

%e .

%e 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 .

%e   +---+---+---+---+

%e   |       |       |

%e   +       +       +

%e   |       |       |

%e   +---+---+       +

%e   |       |       |

%e   +---+---+---+---+

%e   |               |

%e   +---+---+---+---+

%e .

%e 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 Cf. A360499, A360498, A360725, A360256, A360773, A182275, A004003.

%K nonn,more

%O 1,3

%A _Scott R. Shannon_, Feb 21 2023