A368254 Table read by antidiagonals downward: 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

1,2

Links

Table of n, a(n) for n=1..45.

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.

Example

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

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])

Crossrefs

Cf. A368218, A368253.

Sequence in context: A338213 A317736 A060006 * A368261 A368263 A123097

Adjacent sequences: A368251 A368252 A368253 * A368255 A368256 A368257

Keyword

nonn,tabl

Author

Peter Kagey, Dec 19 2023

Status

approved