A368263 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to 180-degree rotation by an asymmetric tile.

1, 3, 2, 4, 7, 2, 10, 20, 16, 4, 16, 76, 88, 43, 4, 36, 272, 720, 538, 120, 9, 64, 1072, 5472, 8356, 3280, 382, 10, 136, 4160, 43968, 131464, 105376, 22028, 1236, 22, 256, 16576, 349568, 2099728, 3355456, 1400536, 149800, 4243, 30

Offset: 1

Author

Peter Kagey, Dec 21 2023

Keywords

• nonn
• tabl

Examples

			Table begins:
  n\k| 1   2     3       4        5          6
  ---+----------------------------------------
   1 | 1   3     4      10       16         36
   2 | 2   7    20      76      272       1072
   3 | 2  16    88     720     5472      43968
   4 | 4  43   538    8356   131464    2099728
   5 | 4 120  3280  105376  3355456  107390592
   6 | 9 382 22028 1400536 89489584 5726776672
		

Links

• Peter Kagey, Illustration of T(2,3)=20
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Programs

Mathematica

A368263[n_, m_] := 1/(2n)*(DivisorSum[n, EulerPhi[#]*2^(n*m/#) &] + n*2^(n*m/2)*Which[EvenQ[m], 1, EvenQ[n], 1/2, True, 0])