A326503 Expansion of Sum_{k>=1} x^k * (1 - x^(2*k)) / (1 + x^k + x^(2*k))^2.

1, -1, 1, 3, -4, -1, 8, -5, 1, 4, -10, 3, 14, -8, -4, 11, -16, -1, 20, -12, 8, 10, -22, -5, 21, -14, 1, 24, -28, 4, 32, -21, -10, 16, -32, 3, 38, -20, 14, 20, -40, -8, 44, -30, -4, 22, -46, 11, 57, -21, -16, 42, -52, -1, 40, -40, 20, 28, -58, -12, 62, -32, 8, 43, -56, 10

Offset

1,4

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Claudia Rella, Resurgence, Stokes constants, and arithmetic functions in topological string theory, arXiv:2212.10606 [hep-th], 2022. See pages 33 - 35.

Formula

a(n) = Sum_{d|n, d==1 (mod 3)} d - Sum_{d|n, d==2 (mod 3)} d.

a(n) = A078181(n) - A078182(n).

Multiplicative with a(3^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) if p == 1 (mod 3) and a(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 2 (mod 3). - Amiram Eldar, Nov 28 2023

Mathematica

nmax = 66; CoefficientList[Series[Sum[x^k (1 - x^(2 k))/(1 + x^k + x^(2 k))^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{1}, Mod[#, 3]] &] - DivisorSum[n, # &, MemberQ[{2}, Mod[#, 3]] &], {n, 1, 66}]
f[p_, e_] := If[Mod[p, 3] == 1, (p^(e + 1) - 1)/(p - 1), ((-p)^(e + 1) - 1)/(-p - 1)]; f[3, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 28 2023 *)

Prog

(PARI) a(n)={sumdiv(n, d, d*((d+1)%3-1))} \\ Andrew Howroyd, Sep 12 2019

Crossrefs

Cf. A002129, A050457, A078181, A078182, A078708, A162397 (Moebius transform), A326401.

Sequence in context: A116392 A324559 A174607 * A105578 A378227 A198739

Adjacent sequences: A326500 A326501 A326502 * A326504 A326505 A326506

Keyword

sign,easy,mult

Author

Ilya Gutkovskiy, Sep 12 2019

Status

approved