A367535 The number of ways of tiling the n X n torus up to the symmetries of the square by a tile that is fixed under horizontal reflection but no other symmetries of the square.

1, 16, 3692, 33570410, 5629501212064, 16397105856182791856, 808450637900676611412052288, 664613997892457939442293683754387488, 9021615045252487149405529092893182593313188608, 2008672555323737844427452615613431716686417747867226446336

Offset

1,2

Links

Table of n, a(n) for n=1..10.

Peter Kagey, Illustration of a(2)=16

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-22.

Mathematica

A367535[n_] := 1/(8 n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 4^(n^2/LCM[c, d])]]]] +If[OddQ[n], 0, n^2 (3*2^(n^2 - 2) + 2^(n^2/2)) ] +2*If[EvenQ[n], n/2*DivisorSum[n, Function[c, EulerPhi[c] (4^(n*n/LCM[2, c]) + 4^((n - 2)*n/LCM[2, c]) If[OddQ[c], 2, 4]^(2 n/c))]], n*DivisorSum[n, Function[c, EulerPhi[c] (4^((n - 1)*n/LCM[2, c]) If[OddQ[c], 2, 4]^(n/c))]]] +n*DivisorSum[n, Function[d, EulerPhi[d]*Which[OddQ[d], 0, EvenQ[d], 2^(n^2/d + 1)]]])

Crossrefs

Cf. A255016, A295223, A367524, A367533, A367534.

Sequence in context: A217021 A375121 A087519 * A222917 A060616 A297387

Adjacent sequences: A367532 A367533 A367534 * A367536 A367537 A367538

Keyword

nonn

Author

Peter Kagey, Dec 13 2023

Status

approved