A368258 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal reflections by two tiles that are each fixed under horizontal reflection.

2, 4, 3, 8, 10, 4, 16, 36, 20, 6, 32, 136, 120, 55, 8, 64, 528, 816, 666, 136, 13, 128, 2080, 5984, 9316, 3536, 430, 18, 256, 8256, 45760, 139656, 106912, 23052, 1300, 30, 512, 32896, 357760, 2164240, 3371840, 1415896, 151848, 4435, 46

Offset: 1

Author

Peter Kagey, Dec 21 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 |  4   20    120      816       5984        45760
   4 |  6   55    666     9316     139656      2164240
   5 |  8  136   3536   106912    3371840    107505280
   6 | 13  430  23052  1415896   89751728   5730905440
   7 | 18 1300 151848 19206736 2454791328 314154568000
		

Links

• Peter Kagey, Illustration of T(2,3)=36
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Programs

Mathematica

A368258[n_,m_] := 1/(2n)*(DivisorSum[n, EulerPhi[#]*2^(n*m/#)&] + n*2^(n*m/2)*If[EvenQ[n], 1/2*(2^m + 1), 2^(m/2)])