A368221 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal reflection by two tiles that are each fixed under horizontal reflection.

2, 4, 3, 8, 10, 6, 16, 36, 40, 10, 32, 136, 288, 136, 20, 64, 528, 2176, 2080, 544, 36, 128, 2080, 16896, 32896, 16640, 2080, 72, 256, 8256, 133120, 524800, 526336, 131328, 8320, 136, 512, 32896, 1056768, 8390656, 16793600, 8390656, 1050624, 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    4      8      16        32          64
   2 |  3   10     36     136       528        2080
   3 |  6   40    288    2176     16896      133120
   4 | 10  136   2080   32896    524800     8390656
   5 | 20  544  16640  526336  16793600   537001984
   6 | 36 2080 131328 8390656 536887296 34359869440
		

Links

• Peter Kagey, Illustration of T(3,2)=40
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Crossrefs

Cf. A368218, A368222.

Programs

Mathematica

A368221[n_, m_] := 1/2 (2^(n*m) + If[EvenQ[n], 2^(n*m/2), 2^(m (n + 1)/2)])