A351322 Number T(n,k) of tilings of a 3k X n rectangle with right trominoes.

1, 1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 0, 1, 1, 0, 8, 8, 4, 1, 1, 0, 16, 0, 18, 0, 1, 1, 0, 32, 64, 88, 72, 8, 1, 1, 0, 64, 0, 468, 384, 162, 0, 1, 1, 0, 128, 512, 2672, 8544, 4312, 520, 16, 1, 1, 0, 256, 0, 16072, 76800, 118586, 22656, 1514, 0, 1, 1, 0, 512, 4096, 100064, 1168512, 3403624, 1795360, 204184, 4312, 32, 1

Offset: 0

Author

Gerhard Kirchner, Feb 21 2022

Keywords

• nonn
• tabl

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

Examples

			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
		

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.

Crossrefs

Cf. A077957, A000079, A046984, A084478, A351323, A351324.

Programs

Maxima

See Maxima Code link.