A368304 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by an asymmetric tile.

1, 4, 4, 6, 28, 6, 23, 194, 194, 23, 52, 2196, 7296, 2196, 52, 194, 26524, 350573, 350573, 26524, 194, 586, 351588, 17895736, 67136624, 17895736, 351588, 586, 2131, 4798174, 954495904, 13744131446, 13744131446, 954495904, 4798174, 2131

Offset

1,2

Links

Table of n, a(n) for n=1..36.

Peter Kagey, Illustration of T(2,2)=28

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
  ---+----------------------------------------------------
   1 |   1      4         6            23               52
   2 |   4     28       194          2196            26524
   3 |   6    194      7296        350573         17895736
   4 |  23   2196    350573      67136624      13744131446
   5 |  52  26524  17895736   13744131446   11258999068672
   6 | 194 351588 954495904 2932037300956 9607679419823148

Mathematica

A368304[n_, m_]:=1/(4*n*m) (DivisorSum[n, Function[d, DivisorSum[m, Function[c, EulerPhi[c]EulerPhi[d]4^(m*n/LCM[c, d])]]]]+If[EvenQ[n], n/2*DivisorSum[m, EulerPhi[#](4^(n*m/LCM[2, #])+4^((n-2)*m/LCM[2, #])*4^(2m/#)*Boole[EvenQ[#]])&], n*DivisorSum[m, EulerPhi[#](4^(n*m/#))&, EvenQ]]+If[EvenQ[m], m/2*DivisorSum[n, EulerPhi[#](4^(n*m/LCM[2, #])+4^((m-2)*n/LCM[2, #])*4^(2n/#)*Boole[EvenQ[#]])&], m*DivisorSum[n, EulerPhi[#](4^(m*n/#))&, EvenQ]]+Which[EvenQ[n]&&EvenQ[m], (n*m)/4 (3*2^(n*m)), OddQ[n*m], 0, OddQ[n+m], (n*m)/2 (2^(n*m))])

Crossrefs

Cf. A222188, A368220, A368257, A368302, A368303, A368306, A368308, A184271.

Sequence in context: A158661 A264578 A261146 * A365063 A090531 A350581

Adjacent sequences: A368301 A368302 A368303 * A368305 A368306 A368307

Keyword

nonn,tabl

Author

Peter Kagey, Dec 21 2023

Status

approved