OFFSET
1,5
COMMENTS
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.
There are 6 tiles, all of size 1 X 1, one for each way of joining two sides of the cell.
LINKS
Jack Hanke, The Mosaic Problem - How and Why to do Math for Fun, Youtube video.
FORMULA
T(n,1) = 0 for all n.
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.
EXAMPLE
Triangle begins:
k=1 k=2 k=3 k=4
n=1: 0;
n=2: 0, 1;
n=3: 0, 73, 31998;
n=4: 0, 3960, 10414981, 20334816290;
n=5: 0, 190475, ...
...
For T(3, 2), there are 73 solutions (squares marked with an asterisk can take any of the six different tiles):
.
1. (36 tilings) 2. (36 tilings) 3. (1 tiling)
+---+---+---+ +---+---+---+ +---+---+---+
| | | | | | | | | | | |
| | | * | | * | | | | |---| |
| /|\ | | | | /|\ | | /| |\ |
+---+---+---+ +---+---+---+ +---+---+---+
| \|/ | | | | \|/ | | \| |/ |
| | | * | | * | | | | |---| |
| | | | | | | | | | | |
+---+---+---+ +---+---+---+ +---+---+---+
CROSSREFS
KEYWORD
AUTHOR
Douglas Boffey, Aug 02 2023
STATUS
approved