OFFSET
1,1
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..500
Petros Hadjicostas, Proof of an explicit formula for Bower's CycleBG transform
FORMULA
CycleBG transform of (3, 0, 0, 0, ...)
CycleBG transform T(A) = invMOEBIUS(invEULER(Carlitz(A)) + A(x^2) - A) + A.
Carlitz transform T(A(x)) has g.f. 1/(1-sum(k>0, (-1)^(k+1)*A(x^k))).
a(n) = (1/n)*sum_{d divides n} phi(n/d)*A092297(d) (n>1). - Azuma Seiichi, Oct 25 2014
a(n) = -1+(-1)^n+A000031(n) (n>1). - Azuma Seiichi, Oct 25 2014 [Corrected by Petros Hadjicostas, Feb 16 2018.]
From Petros Hadjicostas, Feb 16 2018: (Start)
General formula for the CycleBG transform: T(A)(x) = A(x) - Sum_{k>=0} A(x^(2k+1)) + Sum_{k>=1} (phi(k)/k)*log(Carlitz(A)(x^k)). For a proof, see the links above. (For this sequence, A(x) = 3*x.)
G.f.: Sum_{n>=1} a(n)*x^n = 3*x - 2*x/(1-x^2) - Sum_{n>=1} (phi(n)/n)*log(1-2*x^n) = 3*x - Sum_{n>=1} (phi(n)/n)*(2*log(1+x^n) + log(1-2*x^n)).
(End)
MATHEMATICA
a[n_] := If[n==1, 3, Sum[EulerPhi[n/d]*(2*(-1)^d+2^d), {d, Divisors[n]}]/n ];
Array[a, 35] (* Jean-François Alcover, Jul 06 2018, after Andrew Howroyd *)
PROG
(PARI) a(n) = if(n==1, 3, sumdiv(n, d, eulerphi(n/d)*(2*(-1)^d + 2^d))/n); \\ Andrew Howroyd, Oct 14 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Christian G. Bower, Apr 29 2005
STATUS
approved