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Number of ways of tiling the n X n torus up to the symmetries of the square by a tile that is fixed under 180-degree rotation.
4

%I #19 Jul 09 2024 08:55:08

%S 1,23,3776,33601130,5629507922944,16397105889110874288,

%T 808450637900797243544928256,664613997892457948377435344457451552,

%U 9021615045252487149406066393257455761827823616,2008672555323737844427452616231411384297679581096869206528

%N Number of ways of tiling the n X n torus up to the symmetries of the square by a tile that is fixed under 180-degree rotation.

%H Peter Kagey, <a href="/A368137/a368137.pdf">Illustration of a(2)=23</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 A368137[n_] := 1/(8 n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 4^(n^2/LCM[c, d])]]]] + n^2*If[EvenQ[n], 19*2^(n^2 - 2) + 2^(n^2/2), 2^(n^2 + 1)] + n*If[EvenQ[n], DivisorSum[n, Function[d, EulerPhi[ d] (If[EvenQ[d], 2 (2^(n^2/d) + 4^(n^2/d)), 2^(n^2/d)])]], DivisorSum[n, Function[d, EulerPhi[d] (If[EvenQ[d], 2 (2^(n^2/d) + 4^(n^2/d)), 0])]]])

%Y Cf. A255016, A295223, A367524, A367533, A367534, A367535, A367536, A368138.

%K nonn

%O 1,2

%A _Peter Kagey_, Dec 16 2023