A123331 Expansion of (c(q)^2/(3c(q^2))-1)/2 in powers of q where c(q) is a cubic AGM function.

1, 2, 1, 1, 0, 2, 2, 2, 1, 0, 0, 1, 2, 4, 0, 1, 0, 2, 2, 0, 2, 0, 0, 2, 1, 4, 1, 2, 0, 0, 2, 2, 0, 0, 0, 1, 2, 4, 2, 0, 0, 4, 2, 0, 0, 0, 0, 1, 3, 2, 0, 2, 0, 2, 0, 4, 2, 0, 0, 0, 2, 4, 2, 1, 0, 0, 2, 0, 0, 0, 0, 2, 2, 4, 1, 2, 0, 4, 2, 0, 1, 0, 0, 2, 0, 4, 0, 0, 0, 0, 4, 0, 2, 0, 0, 2, 2, 6, 0, 1, 0, 0, 2, 4, 0

Offset: 1

Author

Michael Somos, Sep 26 2006

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A123330(n)=2*a(n) if n>0. A113974(n)=-(-1)^n*a(n).

Cf. A248897.

Programs

Mathematica

f[p_, e_] := If[Mod[p, 6] == 1, e+1, (1+(-1)^e)/2]; f[2, e_] := (3-(-1)^e)/2; f[3, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 22 2023 *)

PARI

{a(n)=if(n<1, 0, -sumdiv(n, d, (-1)^d*kronecker(-3,d)))}

PARI

{a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, (3-(-1)^e)/2, if(p==3, 1, if(p%6==1, e+1, !(e%2)))))))}

PARI

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^3+A)^6/ eta(x+A)^2/eta(x^6+A)^3-1)/2, n))}

Formula

Moebius transform is period 6 sequence [ 1, 1, 0, -1, -1, 0, ...].

a(n) is multiplicative with a(2^e) = (3-(-1)^e)/2, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).

a(3n) = a(4n) = a(n). a(6n+5) = 0.

G.f.: Sum_{k>0} x^k/(1-x^k+x^(2k)) = (theta_3(-q^3)^3/theta_3(-q) - 1)/2.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Nov 14 2023