OFFSET
1,3
COMMENTS
Compare the o.g.f. of this sequence to the following identity:
Sum_{n>=1} -moebius(3*n)*x^n/(1-x^n) = Sum_{n>=0} x^(3^n).
Here phi(n) = A000010(n), the Euler totient function of n.
a(n) = b(n)*c(n) where b(n) = 1, 1, 3, 2, 1,.. is a multiplicative function with b(p^e) = p^(e-1) for p<>3 and p(3^e)=3^e, and where c(n) = 1, 1, 1, 1, 4, 1, 6, 1, 1... is a multiplicative function with c(p^e)=p-1 for p <> 3 and c(3^e)=1. - R. J. Mathar, Jul 02 2013
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..1000
M. Picquet, Applications de la représentation des courbes du troisième degré, Journal de l'École Polytechnique, Paris, 35 (1884), pp. 31-100.
FORMULA
O.g.f.: Sum_{n>=1} -moebius(3*n)*x^n/(1-x^n)^2 = Sum_{n>=1} phi(3*n)/2*x^n.
a(n) = n * Product_{p | n, p prime, p != 3} (1 - 1/p). [Picquet, p. 73.]
a(n) = phi(n)/2*(((2*n^2+1) mod 3)+2). - Gary Detlefs, Dec 06 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = 27/(8*Pi^2) = 0.341958... . - Amiram Eldar, Nov 18 2022
Dirichlet g.f.: zeta(s-1)/(zeta(s)*(1-1/3^s)). - Amiram Eldar, Dec 26 2022
EXAMPLE
G.f.: A(x) = x + x^2 + 3*x^3 + 2*x^4 + 4*x^5 + 3*x^6 + 6*x^7 + 4*x^8 +...
where A(x) = x/(1-x)^2 - x^2/(1-x^2)^2 + 0*x^3/(1-x^3)^2 + 0*x^4/(1-x^4)^2 - x^5/(1-x^5)^2 + 0*x^6/(1-x^6)^2 - x^7/(1-x^7)^2 + 0*x^8/(1-x^8)^2 + 0*x^9/(1-x^9)^2 + x^10/(1-x^10)^2 - x^11/(1-x^11)^2 +...+ -moebius(3*n)*x^n/(1-x^n)^2 +...
MATHEMATICA
EulerPhi[3*Range[100]]/2 (* Harvey P. Dale, Mar 08 2022 *)
PROG
(PARI) {a(n)=polcoeff(sum(m=1, n, -moebius(3*m)*x^m/(1-x^m+x*O(x^n))^2), n)}
(PARI) a(n) = eulerphi(3*n)/2; \\ Michel Marcus, Jun 07 2020
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Paul D. Hanna, Sep 18 2011
EXTENSIONS
Picquet formula and reference added by N. J. A. Sloane, Nov 23 2011
STATUS
approved