Simple Conjecture

[Ralf Stephan]

> I would like to know if the conjecture made in the
> comment, below, has already been proven or at least has
> a name. 
>...
> Apparently (not sure if this is a conjecture or has been proven 
> by someone else) if (z(n)) is any sequence of integers (not all zero) 
> satisfying the forumla z(n) = 2(z(n-2) - z(n-1)) + z(n-3) then 
> |z(n+1)/z(n)| -> golden ratio phi + 1 = (3+sqrt(5))/2 

This is easy to prove. Such sequences have the g.f. P(x)/(1+2x-2x^2-x^3)
and the denominator has the factors 1-x and 1+3x+x^2. Thus, the closed
form involves a constant term plus sequences satisfying
a(n+2)=-3a(n+1)-a(n) which are of form c*r^n+d*s^n with r,s the roots
of 1+3x+x^2.

The C-finite universe is quite boring after time, you can even churn
out the closed forms automagically with a little help of LLL.

Regards,
ralf