A368259 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal reflection by an asymmetric tile.

1, 2, 2, 4, 6, 2, 8, 20, 12, 4, 16, 72, 88, 39, 4, 32, 272, 688, 538, 104, 9, 64, 1056, 5472, 8292, 3280, 366, 10, 128, 4160, 43712, 131464, 104864, 22028, 1172, 22, 256, 16512, 349568, 2098704, 3355456, 1399512, 149800, 4179, 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   2     4       8       16         32
   2 | 2   6    20      72      272       1056
   3 | 2  12    88     688     5472      43712
   4 | 4  39   538    8292   131464    2098704
   5 | 4 104  3280  104864  3355456  107374208
   6 | 9 366 22028 1399512 89489584 5726711136
		

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

A368259[n_,m_]:=1/(2n) (DivisorSum[n,EulerPhi[#]*2^(n*m/#)&]+n*2^(n*m/2-1)*Boole[EvenQ[n]])