OFFSET
0,9
COMMENTS
The table is read by descending antidiagonals.
If read by columns or rows:
T(n,1) = A077957(n+1)
T(2,k) = A000079(k) = 2^k
T(4,k) = A046984(k)
T(5,k) = A084478(k)
T(n,2) = A351323(n)
T(7,k) = A351324(k)
Linear recurrences with different numbers of parameters are known for the sequences above.
Overview:
Constant Number of
side length Sequence parameters
2 T(2,k) 1
3 T(n,1),T(3,k) 2
4 T(4,k) 3 see A046984
5 T(5,k) 4 see A084478
6 T(n,2),T(6,k) 11 see A351323
7 T(7,k) 17 see A351324
8 T(8,k) >30
9 T(n,3),T(9,k) >30
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..495 (first 31 antidiagonals).
Gerhard Kirchner, Tiling algorithm
Gerhard Kirchner, Maxima Code
Gerhard Kirchner, More sequences
Cristopher Moore, Some Polyomino Tilings of the Plane, arXiv:math/9905012 [math.CO], 1999.
EXAMPLE
6 X 2 rectangle: 4 tilings
___ ___ ___ ___
| _| | _| |_ | |_ |
|_| | |_| | | |_| | |_|
|___| |___| |___| |___|
| _| |_ | | _| |_ |
|_| | | |_| |_| | | |_|
|___| |___| |___| |___|
.
Table T(n,k) begins:
n\k__0__1______2_________3_____________4
0: 1 1 1 1 1
1: 1 0 0 0 0
2: 1 2 4 8 16
3: 1 0 8 0 64
4: 1 4 18 88 468
5: 1 0 72 384 8544
6: 1 8 162 4312 118586
7: 1 0 520 22656 1795360
8: 1 16 1514 204184 29986082
9: 1 0 4312 1193600 467966840
10: 1 32 13242 9567192 7758809670
11: 1 0 39088 63112256 124693887784
PROG
(Maxima) See Maxima Code link.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gerhard Kirchner, Feb 21 2022
STATUS
approved