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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 horizontal reflection but no other symmetries of the square.
4

%I #15 Jul 08 2024 10:38:26

%S 1,16,3692,33570410,5629501212064,16397105856182791856,

%T 808450637900676611412052288,664613997892457939442293683754387488,

%U 9021615045252487149405529092893182593313188608,2008672555323737844427452615613431716686417747867226446336

%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 horizontal reflection but no other symmetries of the square.

%H Peter Kagey, <a href="/A367535/a367535.pdf">Illustration of a(2)=16</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 A367535[n_] := 1/(8 n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 4^(n^2/LCM[c, d])]]]] +If[OddQ[n], 0, n^2 (3*2^(n^2 - 2) + 2^(n^2/2)) ] +2*If[EvenQ[n], n/2*DivisorSum[n, Function[c, EulerPhi[c] (4^(n*n/LCM[2, c]) + 4^((n - 2)*n/LCM[2, c]) If[OddQ[c], 2, 4]^(2 n/c))]], n*DivisorSum[n, Function[c, EulerPhi[c] (4^((n - 1)*n/LCM[2, c]) If[OddQ[c], 2, 4]^(n/c))]]] +n*DivisorSum[n, Function[d, EulerPhi[d]*Which[OddQ[d], 0, EvenQ[d], 2^(n^2/d + 1)]]])

%Y Cf. A255016, A295223, A367524, A367533, A367534.

%K nonn

%O 1,2

%A _Peter Kagey_, Dec 13 2023