A368256 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 180-degree rotation but not horizontal or vertical reflections.

1, 2, 2, 3, 5, 2, 6, 14, 9, 4, 10, 44, 52, 26, 4, 20, 152, 366, 298, 62, 8, 36, 560, 2800, 4244, 1704, 205, 9, 72, 2144, 22028, 66184, 52740, 11228, 623, 18, 136, 8384, 175296, 1050896, 1679776, 701124, 75412, 2171, 23

Offset

1,2

Links

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

Peter Kagey, Illustration of T(2,3)=14

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   2     3       6         10           20
   2 | 2   5    14      44        152          560
   3 | 2   9    52     366       2800        22028
   4 | 4  26   298    4244      66184      1050896
   5 | 4  62  1704   52740    1679776     53696936
   6 | 8 205 11228  701124   44758448   2863442960
   7 | 9 623 75412 9591666 1227199056 157073688884

Mathematica

A368256[n_, m_] := 1/(4n)*( DivisorSum[n, EulerPhi[#]*2^(n*m/#) &] + n (2^(n*m/2 - 1))*Boole[EvenQ[n]] + If[EvenQ[m], DivisorSum[n, EulerPhi[#]*2^(n*m/LCM[#, 2]) &], DivisorSum[n, EulerPhi[#]*2^(n*m/#) &, EvenQ]] + n*2^(n*m/2)*Which[EvenQ[m], 1, EvenQ[n], 3/2, True, Sqrt[2]])

Crossrefs

Sequence in context: A117918 A302495 A368255 * A185688 A203955 A039638

Adjacent sequences: A368253 A368254 A368255 * A368257 A368258 A368259

Keyword

nonn,tabl

Author

Peter Kagey, Dec 21 2023

Status

approved