A368260 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to vertical reflections by two tiles that are each fixed under vertical reflection.

2, 3, 3, 6, 7, 4, 10, 24, 14, 6, 20, 76, 100, 40, 8, 36, 288, 700, 564, 108, 14, 72, 1072, 5560, 8296, 3384, 362, 20, 136, 4224, 43800, 131856, 104968, 22288, 1182, 36, 272, 16576, 350256, 2098720, 3358736, 1399176, 150972, 4150, 60

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  14   100     700     5560      43800
   4 |  6  40   564    8296   131856    2098720
   5 |  8 108  3384  104968  3358736  107377488
   6 | 14 362 22288 1399176 89505984 5726689312
		

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

A368260[n_, m_] := 1/(2 n) (DivisorSum[n, EulerPhi[#]*2^(n*m/#) &] + If[EvenQ[m], DivisorSum[n,  EulerPhi[#]*2^(n*m/LCM[#, 2]) &], DivisorSum[n,  EulerPhi[#]*2^((n*m - n)/LCM[#, 2])*2^(n/#) &]])