A368221 Table read by antidiagonals downward: 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

1,1

Links

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

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.

Example

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

Mathematica

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

Crossrefs

Cf. A368218, A368222.

Sequence in context: A368264 A026166 A186003 * A077632 A347976 A253722

Adjacent sequences: A368218 A368219 A368220 * A368222 A368223 A368224

Keyword

nonn,tabl

Author

Peter Kagey, Dec 18 2023

Status

approved