A367530 The number of ways of tiling the n X n torus up to matrix transposition by a tile that is asymmetric with respect to matrix transposition.

1, 4, 32, 2081, 671104, 954448620, 5744387279872, 144115188176529540, 14925010118699132241920, 6338253001141163895983922592, 10985355337065420437221545952731136, 77433143050453552574825990883161180320096, 2213872302702432822841084717014014514981767643136

Offset: 1

Author

Peter Kagey, Dec 13 2023

Keywords

• nonn

Comments

The n X n torus is an n X n grid where two grids are considered the same if one can reach the other by cyclic shifting of rows and columns.

Links

• Peter Kagey, Illustration of a(3) = 32
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023. See also J. Int. Seq., (2024) Vol. 27, Art. No. 24.6.1, pp. A-21, A-25.

Crossrefs

Cf. A103488, A255015.

Programs

Mathematica

A367530[n_] := 1/(2n^2) (DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d])]]]] + n*DivisorSum[n, Function[d, EulerPhi[d]*Which[OddQ[d], 0, EvenQ[d], 2^(n^2/(2 d))]]])