A360498 - OEIS

%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