A368139 Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflection of the square by two tiles that are each fixed under both diagonal and antidiagonal reflection.

2, 6, 36, 1282, 340880, 477513804, 2872221202512, 72057600262282324, 7462505061854009276768, 3169126500572875969052992416, 5492677668532714149024993226980288, 38716571525226776302072008065489884436832, 1106936151351216411420647256070432280699273711360

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)=36

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-23.

Mathematica

A368139[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), (7*2^(n^2/2 - 2))] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*If[EvenQ[d], 2^(n^2/(2 d)), 2^((n^2 + n)/(2d))]]])

Crossrefs

Cf. A295223, A367526, A368140, A368141, A368142.

Sequence in context: A374453 A152480 A001660 * A275904 A014052 A014056

Adjacent sequences: A368136 A368137 A368138 * A368140 A368141 A368142

Keyword

nonn

Author

Peter Kagey, Dec 16 2023

Status

approved