[seqfan] A (new) sequence connected with Fibonacci numbers and Golden ratio

[Vladimir Shevelev]

Dear SeqFans,
 
Is it interesting the following sequence:
Consider consecutive decimal approximations of 2^{\phi}, where \phi is the Golden ratio:
x_1=3, x_2=3.0, x_3=3.06, x_4=3.069, x_5=3.0695 etc.
Then a(n) (n>=1) is the maximal number of the first positive Fibonacci numbers which are given by the sequence defined by the recursion:
A_n(0)=1, A_n(1)=1 and, for m>=2, A_n(m)=floor(log_2(x_n^A_n(m-1)+x_n^A_n(m-2))).
The sequence begins with 6,6,...
 
Regards,
Vladimir

 Shevelev Vladimir‎