A368143 Number of ways of tiling the n X n torus up to 90-degree rotations of the square by two tiles that are each fixed under 90-degree rotation of the square.

2, 6, 28, 1171, 337664, 477339616, 2872202032640, 72057595967392816, 7462505059899322983424, 3169126500571074529242309120, 5492677668532710795071526353530880, 38716571525226776289479030777920527620096, 1106936151351216411420552029913564178922327982080

Offset

1,1

Links

Table of n, a(n) for n=1..13.

S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015  and J. Int. Seq. 18 (2015).

Peter Kagey, Illustration of a(3)=28

Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023. See also J. Int. Seq., (2024) Vol. 27, Art. No. 24.6.1, pp. A-21, A-24.

Mathematica

A368143[n_] := 1/(4n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d])]]]] + n^2*If[OddQ[n], 2^((n^2 + 1)/2) + 2^((n^2 + 7)/4), 7*2^((n^2 - 4)/2) + 5*2^(n^2/4)])

Crossrefs

Cf. A047937, A367534, A368144, A368145.

Sequence in context: A326359 A136639 A368003 * A284594 A027109 A348764

Adjacent sequences: A368140 A368141 A368142 * A368144 A368145 A368146

Keyword

nonn,changed

Author

Peter Kagey, Dec 16 2023

Status

approved