A368141 Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflections of the square by a tile that is fixed under only 180-degree rotations.

1, 4, 24, 1154, 337600, 477339020, 2872202028544, 72057595967315028, 7462505059899321934848, 3169126500571074529202043808, 5492677668532710795071525279789056, 38716571525226776289479030777837491607904, 1106936151351216411420552029913564174524281470976

Offset

1,2

Links

Table of n, a(n) for n=1..13.

Peter Kagey, Illustration of a(3)=24

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

A368141[n_] := 1/(4 n^2) (DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d])]]]] + n^2*If[EvenQ[n], (7*2^((n^2 - 4)/2)), 2^((n^2 + 1)/2)] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*If[EvenQ[d], 2^(n^2/(2 d)), 0]]])

Crossrefs

Cf. A367528, A368137, A368139, A368140, A368142.

Sequence in context: A024251 A193484 A024252 * A368144 A303131 A012124

Adjacent sequences: A368138 A368139 A368140 * A368142 A368143 A368144

Keyword

nonn

Author

Peter Kagey, Dec 16 2023

Status

approved