f(x) = ( f([x/2]) + f([x/3]) ) / 2

[Paul D. Hanna]

Seqfans,
     Very interesting note, Max.
> Consider a function f(x) from the set of non-negative integers N to
> the set of real numbers R such that
> for x>=2, f(x) = ( f([x/2]) + f([x/3]) ) / 2 (here [.] denotes the
> "floor" function).
 
Consider just one of Benoit Cloitre's nice sequences: 
 
A083662: a(n)=a([n/2])+a([n/4]), n>0. a(0)=1.  
COMMENT: 
"For n>0, a(n) = F([log(n)/log(2)]+3) where F(k) denotes the k-th
Fibonacci number. 
For n>=3, F(n) appears 2^(n-3) times. 
More generally, if p is an integer>1 and 
a(n)=a([n/p])+a([n/p^2]), n>0, a(0)=1, 
then for n>0, 
a(n) = F([log(n)/log(p)]+3)."
  
Though very different, it is a noteworthy variant. 
     Paul