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Number of ways of tiling the n X n torus up to the symmetries of the square by an asymmetric tile.
4

%I #20 Jul 09 2024 08:55:14

%S 1,154,1864192,2199026796168,188894659314785812480,

%T 1126800533536206914843196839296,

%U 455117248949604553908892209645884928950272,12259964326927110866866776228808161337250421224373748224,21812926725659065797324660502998994022561529591086874194578215566049280

%N Number of ways of tiling the n X n torus up to the symmetries of the square by an asymmetric tile.

%H Dan Davis, <a href="https://doi.org/10.37236/1338">On a Tiling Scheme from M. C. Escher</a>, Electron. J. Combin. 4(2) (1996), #R23.

%H Peter Kagey, <a href="/A368138/a368138.pdf">Illustration of a(2)=154</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, p. A-23.

%t A368138[n_] := 1/(8n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 8^(n^2/LCM[c, d])]]]] + If[EvenQ[n], n^2 (3/4*8^(n^2/2) + 8^(n^2/4)) + n*DivisorSum[n, Function[c, EulerPhi[c] (If[EvenQ[c], 2*8^(n^2/c), 8^(n^2/(2 c))])]], 0] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*If[EvenQ[d], 8^(n^2/(2 d)), 0]]])

%Y Cf. A255016, A295223, A367525, A367533, A367534, A367535, A367536, A368137.

%K nonn

%O 1,2

%A _Peter Kagey_, Dec 16 2023