A368302 Table read by antidiagonals downward: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by a tile that is fixed under horizontal reflections but not vertical reflections.

1, 2, 2, 2, 5, 2, 4, 9, 8, 4, 4, 26, 22, 22, 4, 9, 62, 120, 126, 44, 8, 10, 205, 600, 1267, 592, 135, 9, 22, 623, 3936, 14164, 13600, 3936, 362, 18, 30, 2171, 25556, 181782, 337192, 178366, 25314, 1211, 23, 62, 7429, 177678, 2437726, 8965354, 8980642, 2404372, 176998, 3914, 44

Offset

1,2

Links

Table of n, a(n) for n=1..55.

Peter Kagey, Illustration of T(3,3)=22

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         9
   2 | 2   5    9     26      62       205
   3 | 2   8   22    120     600      3936
   4 | 4  22  126   1267   14164    181782
   5 | 4  44  592  13600  337192   8965354
   6 | 8 135 3936 178366 8980642 477655760

Mathematica

A368302[n_, m_] := 1/(4*n*m) (DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] + n*If[EvenQ[n], 1/2*DivisorSum[m, EulerPhi[#] (2^(n*m/LCM[2, #]) + 2^((n - 2)*m/LCM[2, #])*2^(2 m/#)) &], DivisorSum[m, EulerPhi[#] (2^((n - 1)*m/LCM[2, #])*2^(m/#)) &]] + m*If[EvenQ[m], 1/2*DivisorSum[n, EulerPhi[#] (2^(n*m/LCM[2, #]) + 2^(m*n/#)*Boole[EvenQ[#]]) &], DivisorSum[n, EulerPhi[#]*2^(m*n/#) &, EvenQ]] + n*m*2^(n*m/2)*Which[EvenQ[n] && EvenQ[m], 3/4, OddQ[n*m], 0, OddQ[n + m], 1/2])

Crossrefs

Cf. A222188, A368218, A368254, A368303, A368304, A368305.

Sequence in context: A046053 A368303 A368306 * A368308 A080348 A355198

Adjacent sequences: A368299 A368300 A368301 * A368303 A368304 A368305

Keyword

nonn,tabl

Author

Peter Kagey, Dec 21 2023

Status

approved