A368305 Table read by antidiagonals downward: T(n,k) is the number of tilings of the n X k torus up to horizontal reflections by two tiles that are both fixed under horizontal reflection.

2, 3, 3, 4, 7, 4, 6, 14, 13, 6, 8, 40, 44, 34, 8, 14, 108, 218, 226, 78, 13, 20, 362, 1200, 2386, 1184, 237, 18, 36, 1182, 7700, 27936, 26892, 7700, 687, 30, 60, 4150, 51112, 361244, 674384, 354680, 50628, 2299, 46

Offset

1,1

Links

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

Peter Kagey, Illustration of T(3,3)=44

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 |  2   3    4      6        8        14
   2 |  3   7   14     40      108       362
   3 |  4  13   44    218     1200      7700
   4 |  6  34  226   2386    27936    361244
   5 |  8  78 1184  26892   674384  17920876
   6 | 13 237 7700 354680 17950356 955180432

Mathematica

A368305[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/#))&]/2, DivisorSum[m, EulerPhi[#](2^((n - 1)*m/LCM[2, #])*2^(m/#))&]])

Crossrefs

Cf. A368221, A368258, A368302, A368306.

Sequence in context: A049790 A222188 A368307 * A184271 A269098 A119795

Adjacent sequences: A368302 A368303 A368304 * A368306 A368307 A368308

Keyword

nonn,tabl

Author

Peter Kagey, Dec 21 2023

Status

approved