A368257 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by an asymmetric tile.

1, 6, 4, 16, 44, 6, 72, 544, 366, 23, 256, 8384, 21856, 4244, 52, 1056, 131584, 1399512, 1050128, 52740, 194, 4096, 2100224, 89478656, 268472384, 53687104, 701124, 586, 16512, 33562624, 5726711136, 68719870208, 54975896016, 2863399264, 9591666, 2131

Offset: 1

Author

Peter Kagey, Dec 21 2023

Keywords

• nonn
• tabl

Examples

			Table begins:
  n\k|   1      2          3              4                 5
  ---+-------------------------------------------------------
   1 |   1      6         16             72               256
   2 |   4     44        544           8384            131584
   3 |   6    366      21856        1399512          89478656
   4 |  23   4244    1050128      268472384       68719870208
   5 |  52  52740   53687104    54975896016    56294995342336
   6 | 194 701124 2863399264 11728132423744 48038396383286784
		

Links

• Peter Kagey, Illustration of T(2,2)=44
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Programs

Mathematica

A368257[n_, m_] := 1/(4n)*(DivisorSum[n, EulerPhi[#]*4^(n*m/#) &] + n (2^(n*m - 1))*Boole[EvenQ[n]] + If[EvenQ[m], DivisorSum[n, EulerPhi[#]*4^(n*m/LCM[#, 2]) &], DivisorSum[n, EulerPhi[#]*4^(n*m/#) &, EvenQ]] + n*2^(n*m)*Which[EvenQ[m], 1, EvenQ[n], 1/2, True, 0])