[seqfan] Sequences appoximating Fibonacci numbers

[Vladimir Shevelev]

Dear SeqFans,
 
Calculating sequence a(0)=1, a(1)=1 and, for n>=2,
 a(n)=floor(log_2(3^a(n-1)+3^a(n-2))),
I (by handy) get: 1,1,2,3,5,8,12,...
>From the arguments of the continuity, there exists a constant x, such that, for Fibonacci sequence {F_n}, we have
F(n)=floor(log_2(x^F(n-1)+x^F(n-2))).
Sequence b(0)=1, b(1)=1 and, for n>=2, 
 b(n)=floor(log_2(\pi^b(n-1)+\pi^b(n-2))), is
1,1,2,3,5,8,13,21,34,56,...
Thus 3<x<\pi.
Can anyone  calculate x with some true signs?
 
Regards,
Vladimir
 

 Shevelev Vladimir‎