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Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflection of the square by an asymmetric tile.
5

%I #16 Jul 09 2024 08:55:51

%S 1,23,7296,67124308,11258999068672,32794211700912270688,

%T 1616901275801313012113145856,1329227995784915876578744356684451904,

%U 18043230090504974298810923860695296894480941056,4017345110647475688854905231100098373350012274109805442048

%N Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflection of the square by an asymmetric tile.

%H Peter Kagey, <a href="/A368142/a368142.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, pp. A-21, A-24.

%t A368142[n_] := 1/(4 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], (3*2^(n^2 - 2)), 0] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*If[EvenQ[d], 2^(n^2/d), 0]]])

%Y Cf. A367529, A368138, A368139, A368140, A368141.

%K nonn

%O 1,2

%A _Peter Kagey_, Dec 16 2023