A368223 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to 180-degree rotation by two tiles that are each fixed under 180-degree rotation.

2, 3, 3, 6, 10, 6, 10, 36, 36, 10, 20, 136, 272, 136, 20, 36, 528, 2080, 2080, 528, 36, 72, 2080, 16512, 32896, 16512, 2080, 72, 136, 8256, 131328, 524800, 524800, 131328, 8256, 136, 272, 32896, 1049600, 8390656, 16781312, 8390656, 1049600, 32896, 272

Offset: 1

Author

Peter Kagey, Dec 18 2023

Keywords

• nonn
• tabl

Examples

			Table begins:
  n\k|  1    2      3       4         5           6
  ---+---------------------------------------------
   1 |  2    3      6      10        20          36
   2 |  3   10     36     136       528        2080
   3 |  6   36    272    2080     16512      131328
   4 | 10  136   2080   32896    524800     8390656
   5 | 20  528  16512  524800  16781312   536887296
   6 | 36 2080 131328 8390656 536887296 34359869440
		

Links

• Peter Kagey, Illustration of T(3,3)=272
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Crossrefs

Cf. A368219, A368224.

Programs

Mathematica

A368223[n_, m_] := 1/2 (2^(n*m) + If[EvenQ[n*m], 2^(n*m/2), 2^((n*m + 1)/2)])