A367533 The number of ways of tiling the n X n torus up to the symmetries of the square by a tile that is fixed under both horizontal and vertical reflection, but not diagonal reflection.

1, 4, 18, 733, 170440, 239035502, 1436110601256, 36028815364865610, 3731252530927004638384, 1584563250299991004659308752, 2746338834266357074512496613490144, 19358285762613388346374725943958077888688, 553468075675608205710323628035216140349636855680

Offset

1,2

Links

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

Peter Kagey, Illustration of a(3)=18

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-22.

Mathematica

A367533[n_] := 1/(8 n^2) * (DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d])]]]] + If[OddQ[n], n^2*2^((n^2 + 1)/2), n^2/4 (3*2^(n^2/2) + 2^((n^2 + 4)/2))] + 2*If[EvenQ[n], n/2*DivisorSum[n, Function[c, EulerPhi[c] (2^(n*n/LCM[2, c]) + 2^((n - 2)*n/LCM[2, c])*2^(2 n/c))]], n*DivisorSum[n, Function[c, EulerPhi[c] (2^((n - 1)*n/LCM[2, c])*2^(n/c))]]] + 2*If[OddQ[n], 0, n^2 2^(n^2/4 - 1)] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*Which[OddQ[d], 0, EvenQ[d], 2^(n^2/(2d))]]])

Crossrefs

Cf. A255016, A295223, A367522.

Sequence in context: A295223 A173222 A342467 * A156491 A287677 A069874

Adjacent sequences: A367530 A367531 A367532 * A367534 A367535 A367536

Keyword

nonn

Author

Peter Kagey, Dec 13 2023

Status

approved