A368255 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 vertical reflections but not horizontal reflections.

1, 2, 2, 3, 5, 2, 6, 14, 9, 4, 10, 44, 50, 26, 4, 20, 152, 366, 298, 62, 9, 36, 560, 2780, 4244, 1692, 205, 10, 72, 2144, 22028, 66184, 52740, 11272, 623, 22, 136, 8384, 175128, 1050896, 1679368, 701124, 75486, 2171, 30

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    50     366       2780        22028
   4 |  4  26   298    4244      66184      1050896
   5 |  4  62  1692   52740    1679368     53696936
   6 |  9 205 11272  701124   44761184   2863442960
   7 | 10 623 75486 9591666 1227208420 157073688884

Mathematica

A368255[n_, m_] := 1/(4n)*(DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)]] + n*(2^(n*m/2 - 1))*Boole[EvenQ[n]] + If[EvenQ[m], DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/LCM[d, 2])]], DivisorSum[n, Function[d, EulerPhi[d]*2^((n*m - n)/LCM[d, 2])*2^(n/d)]]] + n*2^(n*m/2)*Which[EvenQ[m], 1, EvenQ[n], 1/2, True, 0])

Crossrefs

Cf. A368218, A368253, A368254, A368256, A368257, A368260.

Sequence in context: A297495 A117918 A302495 * A368256 A185688 A203955

Adjacent sequences: A368252 A368253 A368254 * A368256 A368257 A368258

Keyword

nonn,tabl

Author

Peter Kagey, Dec 21 2023

Status

approved