A368305 -id:A368305 - cp's OEIS frontend

A368302 Table read by downward antidiagonals: 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

Peter Kagey, Dec 21 2023

			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

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.

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

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])

A368306 Table read by downward antidiagonals: 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.

Original entry on oeis.org

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

Peter Kagey, Dec 21 2023

			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

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.

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

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]])

cp's OEIS Frontend

A368302 Table read by downward antidiagonals: 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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica