OFFSET
0,9
LINKS
Pontus von Brömssen, Table of n, a(n) for n = 0..90 (rows 0..12)
EXAMPLE
Triangle begins:
n\k| 0 1 2 3 4 5 6 7 8
---+-----------------------------------------
0 | 1
1 | 1 1
2 | 1 1 1
3 | 1 1 2 4
4 | 1 1 3 6 16
5 | 1 1 4 12 37 140
6 | 1 1 6 24 105 454 1987
7 | 1 1 10 40 250 1566 9856 62266
8 | 1 1 15 80 726 5670 47394 406168 3899340
A 5 X 4 rectangle can be tiled by 12 unit squares and 2 squares of side 2 in the following ways:
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | | | | | | | |
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | | | | |
+---+---+---+---+ + +---+---+ +---+ +---+ +---+---+ +
| | | | | | | | | | | | | | | |
+---+---+ + +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | | |
+ +---+---+ + +---+---+ + +---+---+ + +---+---+
| | | | | | | | | | | | | | | |
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
.
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | | | | | |
+---+ +---+ +---+---+---+---+ +---+---+---+---+ + +---+---+
| | | | | | | | | | | | | | | | |
+---+---+---+---+ + +---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | | | | | | |
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | | |
+ +---+---+ +---+ +---+ + + + + +---+---+
| | | | | | | | | | | | | | |
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
.
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | | | | | | |
+---+---+ + +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | | | | |
+---+---+---+---+ +---+ +---+ +---+---+---+---+ +---+---+ +
| | | | | | | | | | | | | | |
+---+---+---+---+ +---+---+---+---+ + + + + +---+---+
| | | | | | | | | | | | | | |
+ +---+---+ +---+ +---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | | | | | | | | |
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
.
+---+---+---+---+
| | | |
+---+ +---+
| | | |
+---+---+---+---+
| | | | |
+---+---+---+---+
| | | |
+---+ +---+
| | | |
+---+---+---+---+
The first six of these have no symmetries, so they account for 4 tilings each. The next six have either a mirror symmetry or a rotational symmetry and account for 2 tilings each. The last has full symmetry and accounts for 1 tiling. In total there are 6*4+6*2+1 = 37 tilings. This is the maximum for a 5 X 4 rectangle, so T(5,4) = 37.
CROSSREFS
Main diagonal: A362143.
KEYWORD
nonn,tabl
AUTHOR
Pontus von Brömssen, Apr 10 2023
STATUS
approved