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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.
6

%I #16 Jul 08 2024 10:38:20

%S 1,4,18,733,170440,239035502,1436110601256,36028815364865610,

%T 3731252530927004638384,1584563250299991004659308752,

%U 2746338834266357074512496613490144,19358285762613388346374725943958077888688,553468075675608205710323628035216140349636855680

%N 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.

%H Peter Kagey, <a href="/A367533/a367533.pdf">Illustration of a(3)=18</a>

%H Peter Kagey and William Keehn, <a href="https://arxiv.org/abs/2311.13072">Counting tilings of the n X m grid, cylinder, and torus</a>, arXiv: 2311.13072 [math.CO], 2023. See also <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Kagey/kagey6.html">J. Int. Seq.</a>, (2024) Vol. 27, Art. No. 24.6.1, pp. A-21, A-22.

%t 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))]]])

%Y Cf. A255016, A295223, A367522.

%K nonn

%O 1,2

%A _Peter Kagey_, Dec 13 2023