A368261 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to vertical reflection by an asymmetric tile.

1, 3, 2, 4, 7, 2, 10, 20, 14, 4, 16, 76, 88, 40, 4, 36, 272, 700, 532, 108, 8, 64, 1072, 5472, 8296, 3280, 362, 10, 136, 4160, 43800, 131344, 104968, 21944, 1182, 20, 256, 16576, 349568, 2098720, 3355456, 1399176, 149800, 4150, 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  14    88     700     5472      43800
   4 | 4  40   532    8296   131344    2098720
   5 | 4 108  3280  104968  3355456  107377488
   6 | 8 362 21944 1399176 89484128 5726689312
		

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

A368261[n_, m_]:=1/(2n)*(DivisorSum[n, EulerPhi[#]*2^(n*m/#)&] + If[EvenQ[m], DivisorSum[n, EulerPhi[#]*2^(n*m/LCM[#, 2])&], DivisorSum[n, EulerPhi[#]*2^(n*m/#)&, EvenQ]])