A368142 Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflection of the square by an asymmetric tile.

1, 23, 7296, 67124308, 11258999068672, 32794211700912270688, 1616901275801313012113145856, 1329227995784915876578744356684451904, 18043230090504974298810923860695296894480941056, 4017345110647475688854905231100098373350012274109805442048

Offset

1,2

Links

Table of n, a(n) for n=1..10.

Peter Kagey, Illustration of a(2)=23

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-24.

Mathematica

A368142[n_] := 1/(4 n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 4^(n^2/LCM[c, d])]]]] + n^2*If[EvenQ[n], (3*2^(n^2 - 2)), 0] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*If[EvenQ[d], 2^(n^2/d), 0]]])

Crossrefs

Cf. A367529, A368138, A368139, A368140, A368141.

Sequence in context: A233109 A193430 A233213 * A368145 A090674 A013728

Adjacent sequences: A368139 A368140 A368141 * A368143 A368144 A368145

Keyword

nonn

Author

Peter Kagey, Dec 16 2023

Status

approved