A368222 Table read by antidiagonals downward: T(n,k) is the number of tilings of the n X k grid up to horizontal reflection by an asymmetric tile.

1, 2, 3, 4, 10, 4, 8, 36, 32, 10, 16, 136, 256, 136, 16, 32, 528, 2048, 2080, 512, 36, 64, 2080, 16384, 32896, 16384, 2080, 64, 128, 8256, 131072, 524800, 524288, 131328, 8192, 136, 256, 32896, 1048576, 8390656, 16777216, 8390656, 1048576, 32896, 256

Offset

1,2

Links

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

Peter Kagey, Illustration of T(3,2)=32

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 |  1    2      4       8        16          32
   2 |  3   10     36     136       528        2080
   3 |  4   32    256    2048     16384      131072
   4 | 10  136   2080   32896    524800     8390656
   5 | 16  512  16384  524288  16777216   536870912
   6 | 36 2080 131328 8390656 536887296 34359869440

Mathematica

A368222[n_, m_] := 2^(n*m/2 - 1) (2^(n*m/2) + Boole[EvenQ[n]])

Crossrefs

Cf. A368220, A368221, A368224, A117401.

Sequence in context: A286709 A082865 A359996 * A265348 A158929 A255479

Adjacent sequences: A368219 A368220 A368221 * A368223 A368224 A368225

Keyword

nonn,tabl

Author

Peter Kagey, Dec 18 2023

Status

approved