A368140 Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflection of the square by a tile that is fixed under only diagonal reflection.

1, 4, 22, 1154, 337192, 477360876, 2872203226920, 72057597041056852, 7462505060326909791920, 3169126500571693774150807456, 5492677668532711895587506949961184, 38716571525226776294594927800946276718944, 1106936151351216411420589971585441310578379941760

Offset: 1

Author

Peter Kagey, Dec 16 2023

Keywords

• nonn

Links

• Peter Kagey, Illustration of a(3)=22
• 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.

Crossrefs

Cf. A295223, A367526, A368139, A368141, A368142.

Programs

Mathematica

A368140[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[EvenQ[n], (3*2^(n^2/2 - 2)), 0] + n*DivisorSum[n, Function[d, EulerPhi[d] If[EvenQ[d], 2^(n^2/(2 d) + 1), 2^((n^2 + n)/(2d))]]])