OFFSET
1,5
COMMENTS
The sequence 1,1,1,1,2,3,4,5,8,13,18,25,40,62,90,135,... appears in Lehrer-Segal on p. 285, in the following context: Let V=Sum_{k>=1} V_k be the graded vector space H_*(PC^oo)[1], which has Poincaré series [or Poincare series] p(t)=t/(1-t^2). This sequence gives the dimensions of the free graded Lie algebra L on V.
Inverse Euler transform of F(1-n) where F() is Fibonacci numbers A000045. - Michael Somos, Jul 21 2003
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..2000
G. I. Lehrer and G. B. Segal, Homology stability for classical regular semisimple varieties, Math. Zeit., 236 (2001), 251-290.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
N. J. A. Sloane, Transforms
FORMULA
a(n) = (1/n)*Sum_{d|n} (-1)^d*mu(n/d)*(Fibonacci(d-1)+Fibonacci(d+1)-1). - Vladeta Jovovic, May 03 2003
a(n) ~ (-1)^n * phi^n / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 09 2019
MATHEMATICA
a[n_] := DivisorSum[n, (-1)^#*MoebiusMu[n/#]*(Fibonacci[#+1] + Fibonacci[# -1]-1)&]/n; Array[a, 40] (* Jean-François Alcover, Dec 03 2015, adapted from PARI *)
PROG
(PARI) a(n)=if(n<1, 0, sumdiv(n, d, (-1)^d*moebius(n/d)*(fibonacci(d+1)+fibonacci(d-1)-1))/n)
CROSSREFS
KEYWORD
sign
AUTHOR
N. J. A. Sloane, Nov 19 2001
EXTENSIONS
More terms and formula from Christian G. Bower, Aug 23 2002
STATUS
approved