A368137 - OEIS

A368137

Number of ways of tiling the n X n torus up to the symmetries of the square by a tile that is fixed under 180-degree rotation.

1, 23, 3776, 33601130, 5629507922944, 16397105889110874288, 808450637900797243544928256, 664613997892457948377435344457451552, 9021615045252487149406066393257455761827823616, 2008672555323737844427452616231411384297679581096869206528

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OFFSET

1,2

LINKS

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

Peter Kagey, Illustration of a(2)=23

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, p. A-23.

MATHEMATICA

A368137[n_] := 1/(8 n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 4^(n^2/LCM[c, d])]]]] + n^2*If[EvenQ[n], 19*2^(n^2 - 2) + 2^(n^2/2), 2^(n^2 + 1)] + n*If[EvenQ[n], DivisorSum[n, Function[d, EulerPhi[ d] (If[EvenQ[d], 2 (2^(n^2/d) + 4^(n^2/d)), 2^(n^2/d)])]], DivisorSum[n, Function[d, EulerPhi[d] (If[EvenQ[d], 2 (2^(n^2/d) + 4^(n^2/d)), 0])]]])

CROSSREFS

Cf. A255016, A295223, A367524, A367533, A367534, A367535, A367536, A368138.

Sequence in context: A222031 A233143 A134798 * A352977 A308458 A103443

Adjacent sequences: A368134 A368135 A368136 * A368138 A368139 A368140

KEYWORD

nonn

AUTHOR

Peter Kagey, Dec 16 2023

STATUS

approved