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

1,1

Links

Table of n, a(n) for n=1..45.

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.

Example

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

Mathematica

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

Crossrefs

Cf. A368219, A368224.

Sequence in context: A129068 A079888 A165257 * A059191 A124063 A054630

Adjacent sequences: A368220 A368221 A368222 * A368224 A368225 A368226

Keyword

nonn,tabl

Author

Peter Kagey, Dec 18 2023

Status

approved