A368306 Table read by antidiagonals downward: T(n,k) is the number of tilings of the n X k torus up to horizontal reflections by a tile that is not fixed under horizontal reflection.

1, 2, 2, 2, 5, 2, 4, 8, 9, 4, 4, 24, 32, 26, 4, 8, 56, 186, 182, 62, 9, 10, 190, 1096, 2130, 1096, 205, 10, 20, 596, 7356, 26296, 26380, 7356, 623, 22, 30, 2102, 49940, 350316, 671104, 350584, 49940, 2171, 30

Offset

1,2

Links

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

Peter Kagey, Illustration of T(3,3)=32

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    2      4        4         8
   2 | 2   5    8     24       56       190
   3 | 2   9   32    186     1096      7356
   4 | 4  26  182   2130    26296    350316
   5 | 4  62 1096  26380   671104  17899020
   6 | 9 205 7356 350584 17897924 954481360

Mathematica

A368306[n_, m_] := 1/(2*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] + n*If[EvenQ[n], DivisorSum[m, EulerPhi[#] (2^(n*m/LCM[2, #]) + 2^((n - 2)*m/LCM[2, #])*4^(m/#)*Boole[EvenQ[#]]) &]/2, DivisorSum[m, EulerPhi[#]*2^(n*m/#) &, EvenQ]])

Crossrefs

Cf. A368222, A368259, A368304, A368305, A368308, A184271.

Sequence in context: A095370 A046053 A368303 * A368302 A368308 A080348

Adjacent sequences: A368303 A368304 A368305 * A368307 A368308 A368309

Keyword

nonn,tabl

Author

Peter Kagey, Dec 21 2023

Status

approved