%I #18 Jul 09 2024 08:55:26
%S 1,17,3692,33572458,5629501212064,16397105857614447792,
%T 808450637900676611412052288,664613997892457939730524059906099232,
%U 9021615045252487149405529092893182593313188608,2008672555323737844427452615629277349189270615385935288832
%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 matrix transposition but no other symmetries.
%C A Truchet tile is an example of a tile that is fixed under matrix transposition but no other symmetries.
%H Peter Kagey, <a href="/A367536/a367536.pdf">Illustration of a(2)=17</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-23.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TruchetTiling.html">Truchet Tiling</a>
%t A367536[n_] := 1/(8n^2) (DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 4^(n^2/LCM[c, d])]]]] +If[OddQ[n],n*DivisorSum[n, Function[c, EulerPhi[c] 2^(n^2/c + 1)]],n*DivisorSum[n, Function[c, EulerPhi[c] (4^(n^2/LCM[2, c]) + 2^(n^2/c + 1) + If[OddQ[c], 0, 4^(n^2/c)])]] + n^2 (3*2^(n^2 - 2) + 2^(n^2/2))])
%Y Cf. A255016, A295223, A302484, A367533, A367534, A367535.
%K nonn
%O 1,2
%A _Peter Kagey_, Dec 13 2023