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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 90-degree rotation but not reflection.
6

%I #14 Jul 08 2024 10:38:23

%S 1,4,14,613,168832,238686222,1436101016320,36028798185029194,

%T 3731252529949661491712,1584563250285579485868500176,

%U 2746338834266355397535763176765440,19358285762613388144887089341554236250288,553468075675608205710276014956782089461163991040

%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 90-degree rotation but not reflection.

%H Peter Kagey, <a href="/A367534/a367534.pdf">Illustration of a(3)=14</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 A367534[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]) If[OddQ[c], 0, 2^(2 n/c)])]], n*DivisorSum[n, Function[c, EulerPhi[ c] (2^((n - 1)*n/LCM[2, c]) If[OddQ[c], 0, 2^(n/c)])]]] + 2*If[OddQ[n], n^2 2^((n^2 + 3)/4), n^2/2 (2^(n^2/4) + 2^(n^2/4 + 2))] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*Which[OddQ[d], 0, EvenQ[d], 2^(n^2/(2d))]]])

%Y Cf. A255016, A295223, A367523, A367533.

%K nonn

%O 1,2

%A _Peter Kagey_, Dec 13 2023