A367534 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 90-degree rotation but not reflection.

1, 4, 14, 613, 168832, 238686222, 1436101016320, 36028798185029194, 3731252529949661491712, 1584563250285579485868500176, 2746338834266355397535763176765440, 19358285762613388144887089341554236250288, 553468075675608205710276014956782089461163991040

Offset

1,2

Links

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

Peter Kagey, Illustration of a(3)=14

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

A367534[n_] := 1/(8 n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d])]]]] + If[OddQ[n], n^2 2^((n^2 + 1)/2), n^2/4 (3*2^(n^2/2) + 2^((n^2 + 4)/2))] + 2*If[EvenQ[n], n/2*DivisorSum[n, Function[c, EulerPhi[ c] (2^(n*n/LCM[2, c]) + 2^((n - 2)*n/LCM[2, c]) If[OddQ[c], 0, 2^(2 n/c)])]], n*DivisorSum[n, Function[c, EulerPhi[ c] (2^((n - 1)*n/LCM[2, c]) If[OddQ[c], 0, 2^(n/c)])]]] + 2*If[OddQ[n], n^2 2^((n^2 + 3)/4), n^2/2 (2^(n^2/4) + 2^(n^2/4 + 2))] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*Which[OddQ[d], 0, EvenQ[d], 2^(n^2/(2d))]]])

Crossrefs

Cf. A255016, A295223, A367523, A367533.

Sequence in context: A118770 A226943 A292708 * A112514 A001140 A177363

Adjacent sequences: A367531 A367532 A367533 * A367535 A367536 A367537

Keyword

nonn,changed

Author

Peter Kagey, Dec 13 2023

Status

approved