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%I #19 Jul 09 2024 08:56:24
%S 1,4,24,1155,337600,477339104,2872202028544,72057595967327280,
%T 7462505059899321934848,3169126500571074529208754688,
%U 5492677668532710795071525279789056,38716571525226776289479030777851808143360,1106936151351216411420552029913564174524281470976
%N Number of ways of tiling the n X n torus up to 90-degree rotations of the square by a tile that is fixed only under 180-degree rotation of the square.
%H Peter Kagey, <a href="/A368144/a368144.pdf">Illustration of a(3)=24</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-25.
%t A368144[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[OddQ[n], 2^((n^2 + 1)/2), 7/4*2^(n^2/2) + 2^(n^2/4)])
%Y Cf. A367531, A368137, A368141, A368143, A368145.
%K nonn
%O 1,2
%A _Peter Kagey_, Dec 16 2023