A364440 - OEIS

%I #50 Sep 06 2023 20:30:00

%S 0,0,1,0,73,31998,0,3960,10414981,20334816290,0,190475

%N Triangle T(n,k) (n >= 1 and 1 <= k <= n) read by rows, arising from the Mosaic Problem.

%C Fill an n X k array of cells with tiles taken from a set of six (each one connecting two sides of the cell). T(n,k) is the number of tilings containing at least one loop.

%C There are 6 tiles, all of size 1 X 1, one for each way of joining two sides of the cell.

%H Jack Hanke, <a href="https://www.youtube.com/watch?v=D3dp5RBmPcs">The Mosaic Problem - How and Why to do Math for Fun</a>, Youtube video.

%F T(n,1) = 0 for all n.

%F T(n,2) = 36^n - ((36*beta - 35)*beta^(1 - n) - (36*alpha - 35)*alpha^(1 - n))/(beta - alpha), where alpha = (1 + sqrt(33/37))/2 and beta = (1 - sqrt(33/37))/2.

%e Triangle begins:

%e         k=1    k=2       k=3          k=4

%e   n=1:   0;

%e   n=2:   0,      1;

%e   n=3:   0,     73,    31998;

%e   n=4:   0,   3960, 10414981, 20334816290;

%e   n=5:   0, 190475, ...

%e   ...

%e For T(3, 2), there are 73 solutions (squares marked with an asterisk can take any of the six different tiles):

%e .

%e 1. (36 tilings)   2. (36 tilings)   3. (1 tiling)

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

%e   |   |   |   |     |   |   |   |     |   |   |   |

%e   |   |   | * |     | * |   |   |     |   |---|   |

%e   |  /|\  |   |     |   |  /|\  |     |  /|   |\  |

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

%e   |  \|/  |   |     |   |  \|/  |     |  \|   |/  |

%e   |   |   | * |     | * |   |   |     |   |---|   |

%e   |   |   |   |     |   |   |   |     |   |   |   |

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

%K nonn,tabl,hard,more

%O 1,5

%A _Douglas Boffey_, Aug 02 2023