This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A364440 #50 Sep 06 2023 20:30:00 %S A364440 0,0,1,0,73,31998,0,3960,10414981,20334816290,0,190475 %N A364440 Triangle T(n,k) (n >= 1 and 1 <= k <= n) read by rows, arising from the Mosaic Problem. %C A364440 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 A364440 There are 6 tiles, all of size 1 X 1, one for each way of joining two sides of the cell. %H A364440 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 A364440 T(n,1) = 0 for all n. %F A364440 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 A364440 Triangle begins: %e A364440 k=1 k=2 k=3 k=4 %e A364440 n=1: 0; %e A364440 n=2: 0, 1; %e A364440 n=3: 0, 73, 31998; %e A364440 n=4: 0, 3960, 10414981, 20334816290; %e A364440 n=5: 0, 190475, ... %e A364440 ... %e A364440 For T(3, 2), there are 73 solutions (squares marked with an asterisk can take any of the six different tiles): %e A364440 . %e A364440 1. (36 tilings) 2. (36 tilings) 3. (1 tiling) %e A364440 +---+---+---+ +---+---+---+ +---+---+---+ %e A364440 | | | | | | | | | | | | %e A364440 | | | * | | * | | | | |---| | %e A364440 | /|\ | | | | /|\ | | /| |\ | %e A364440 +---+---+---+ +---+---+---+ +---+---+---+ %e A364440 | \|/ | | | | \|/ | | \| |/ | %e A364440 | | | * | | * | | | | |---| | %e A364440 | | | | | | | | | | | | %e A364440 +---+---+---+ +---+---+---+ +---+---+---+ %K A364440 nonn,tabl,hard,more %O A364440 1,5 %A A364440 _Douglas Boffey_, Aug 02 2023