<s>[INST] Does $a_{n}/a_{n-1}$ converge to the golden ratio for all Fibonacci-like sequences? Yesterday a friend challenged me to prove that 
$$\lim_{n\rightarrow\infty}\frac{a_n}{a_{n-1}}=\varphi\; ,$$
where $\varphi$ is the golden ratio, for the Fibonacci series.
I started rewriting the limit as
$$\lim_{n\rightarrow\infty}\frac{a_n}{a_{n-1}}=\lim_{n\rightarrow\infty}\frac{a_{n-1}+a_{n-2}}{a_{n-1}}=\lim_{n\rightarrow\infty}1+\frac{a_{n-2}}{a_{n-1}}\; .$$
If the sequence $b_n=\frac{a_n}{a_{n-1}}$ is convergent,
$$\lim_{n\rightarrow\infty}\frac{a_{n-2}}{a_{n-1}}=\left(\lim_{n\rightarrow\infty}\frac{a_n}{a_{n-1}}\right)^{-1}\; .$$
Renaming the desired limit $x$, we obtain the quadratic equation 
$$x=1+\frac{1}{x}$$
$$x^2-x-1=0$$
if $x\neq 0$. Therefore, if $b_n$ is convergent, it must be equal to $\frac{1+\sqrt{5}}{2}$ or $\frac{1-\sqrt{5}}{2}$.
Since $a_n>0$, $b_n>0, \forall n$, so the limit must be equal to $\varphi=\frac{1+\sqrt{5}}{2}$.
This proof made me think that I didn't make use of the initial values of the sequence, so it must hold true for any sequence where $a_{n}=a_{n-1}+a_{n-2}$. The first question is, is $a_{n}/a_{n-1}$ convergent for all Fibonacci-like sequences?
The second and most intriguing for me is, is there any Fibonacci-like sequence where the limit is $\frac{1-\sqrt{5}}{2}$? Since this solution is negative, $a_n$ should change its sing with each $n$, but I couldn't find any values for $a_0$ and $a_1$, which would lead me to this case. If the answer to this question is no, what mathematical sense does this negative solution have?
 [/INST] One way to look at this problem is that if $a_{n+1}=a_n+a_{n-1}$, then we we have
$$\begin{pmatrix}0&1\\1&1\end{pmatrix}\begin{pmatrix}a_{n-1}\\a_n\end{pmatrix}=\begin{pmatrix}a_n\\a_{n+1}\end{pmatrix}.$$
The eigenvalues of the matrix 
$$\begin{pmatrix}0&1\\1&1\end{pmatrix}$$
are $$\dfrac{1+\sqrt{5}}{2},\dfrac{1-\sqrt{5}}{2}.$$
These have corresponding eigenvectors $v_1,v_2$ which span $\mathbb{R}^2$. This leads us to the conclusion that $$\begin{pmatrix}a_1\\a_2\end{pmatrix}=cv_1+bv_2,$$
and if $c\not=0,$ then $v_1$ will dominate the sequence, and we can show the ratios converge to $(1+\sqrt{5})/2.$ This leads us to the conclusion that if we want a Fibonacci like sequence to have ratios converging to $(1-\sqrt{5})/2$, then we must have
$$\begin{pmatrix}a_1\\a_2\end{pmatrix}=bv_2$$
for some non zero $b\in\mathbb{R}$. So to determine all such sequences we simply have to have an eigenvector $v_2$ corresponding to $(1-\sqrt{5})/2$. One such eigenvector is
$$\begin{pmatrix}1\\\dfrac{1-\sqrt{5}}{2}\end{pmatrix}.$$
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