A368254 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 a tile that is fixed under horizontal reflections but not vertical reflections.

1, 3, 2, 4, 7, 2, 10, 20, 13, 4, 16, 76, 60, 34, 4, 36, 272, 430, 346, 78, 8, 64, 1072, 2992, 4756, 1768, 237, 9, 136, 4160, 23052, 70024, 53764, 11612, 687, 18, 256, 16576, 178880, 1083664, 1685920, 709316, 75924, 2299, 23

Offset: 1

Author

Peter Kagey, Dec 19 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  13    60    430     2992      23052
   4 | 4  34   346   4756    70024    1083664
   5 | 4  78  1768  53764  1685920   53762472
   6 | 8 237 11612 709316 44881328 2865540112
		

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.

Crossrefs

Cf. A368218, A368253.

Programs

Mathematica

A368254[n_, m_] := 1/(4n)(DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)]] + n*2^(n*m/2)*If[EvenQ[n], 1/2 (2^m + 1), 2^(m/2)] + If[EvenQ[m], DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/LCM[d, 2])]], DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)], EvenQ]] + n*2^(n*m/2)*Which[EvenQ[m], 1, EvenQ[n], 1/2, True, 0])