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Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflections of the square by a tile that is fixed under only 180-degree rotations.
5

%I #14 Jul 09 2024 08:55:55

%S 1,4,24,1154,337600,477339020,2872202028544,72057595967315028,

%T 7462505059899321934848,3169126500571074529202043808,

%U 5492677668532710795071525279789056,38716571525226776289479030777837491607904,1106936151351216411420552029913564174524281470976

%N Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflections of the square by a tile that is fixed under only 180-degree rotations.

%H Peter Kagey, <a href="/A368141/a368141.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-24.

%t A368141[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[EvenQ[n], (7*2^((n^2 - 4)/2)), 2^((n^2 + 1)/2)] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*If[EvenQ[d], 2^(n^2/(2 d)), 0]]])

%Y Cf. A367528, A368137, A368139, A368140, A368142.

%K nonn

%O 1,2

%A _Peter Kagey_, Dec 16 2023