login
Number of ways of tiling the n X n torus up to 90-degree rotations of the square by an asymmetric tile.
3

%I #25 Jul 09 2024 08:56:11

%S 1,23,7296,67124336,11258999068672,32794211700912314368,

%T 1616901275801313012113145856,1329227995784915876578744357489750016,

%U 18043230090504974298810923860695296894480941056,4017345110647475688854905231100098373350012499289786810368

%N Number of ways of tiling the n X n torus up to 90-degree rotations of the square by an asymmetric tile.

%C M.C. Escher enumerated a(2) = 23 by hand in May 1942, being perhaps the first person to attempt this sort of counting problem. (See Doris Schattschneider's book in the references for more details.)

%D Doris Schattschneider, Visions of Symmetry, W.H. Freeman, 1990, pages 44-48.

%H Peter Kagey, <a href="/A368145/a368145.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-25.

%H Doris Schattschneider, <a href="https://doi.org/10.37236/1332">Escher's combinatorial patterns</a>, Electron. J. Combin. 4(2) (1996), #R17.

%t A368145[n_] := 1/(4n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 4^(n^2/LCM[c, d])]]]] + n^2*If[OddQ[n], 0, 3/4*2^n^2 + 2^(n^2/2)])

%Y Cf. A367530, A367532, A368138, A368142, A368143, A368144.

%K nonn

%O 1,2

%A _Peter Kagey_, Dec 16 2023