%I #8 Mar 11 2023 23:07:12

%S 0,1,1,0,1,0,1,0,0,4128,1,10880,641,45904,349496,892088,40873,17695080

%N Number of ways to tile an n X n square using rectangles with distinct dimensions where all the rectangle edge lengths are prime numbers.

%C All possible tilings are counted, including those identical by symmetry. Note that distinct dimensions means that, for example, a 2 X 3 rectangle can only be used once, regardless of whether it lies horizontally or vertically.

%e a(2), a(3), a(5), a(7), a(11) = 1 as the only possible tiling is that using an n X n square where n is a prime number. It is likely 11 is the last prime indexed term that equals 1 although this is unknown.

%e a(10) = 4128. And example tiling is:

%e .

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

%e | | | |

%e + + + +

%e | | | |

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

%e | | |

%e + + +

%e | | |

%e + + +

%e | | |

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

%e | | |

%e + + +

%e | | |

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

%e | | |

%e + + +

%e | | |

%e + + +

%e | | |

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

%e .

%Y Cf. A360943, A360499, A360804, A360256, A360773, A182275, A004003.

%K nonn,more

%O 1,10

%A _Scott R. Shannon_, Mar 10 2023