A368262 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 two tiles that are each fixed under 180-degree rotation.

2, 3, 3, 6, 7, 4, 10, 24, 16, 6, 20, 76, 104, 43, 8, 36, 288, 720, 570, 120, 13, 72, 1072, 5600, 8356, 3408, 382, 18, 136, 4224, 43968, 131976, 105376, 22284, 1236, 30, 272, 16576, 350592, 2099728, 3359552, 1400536, 150824, 4243, 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   3     6      10       20         36
   2 |  3   7    24      76      288       1072
   3 |  4  16   104     720     5600      43968
   4 |  6  43   570    8356   131976    2099728
   5 |  8 120  3408  105376  3359552  107390592
   6 | 13 382 22284 1400536 89505968 5726776672
		

Links

• Peter Kagey, Illustration of T(2,3)=24
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Programs

Mathematica

A368262[n_, m_] := 1/(2n)*(DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)]] + n*2^(n*m/2)*Which[EvenQ[m], 1, EvenQ[n], 3/2, True, Sqrt[2]])