It is well known that for each pair $f_n,f_m$ in Fibonacci sequence we have $$\gcd(f_n,f_m)=f_{\gcd(n,m)}$$
What about the other way? If we have $\gcd(a_n,a_m)=a_{\gcd(n,m)}$ for each pair in nonconstant sequence $a_1,a_2,...$ of natural nubers, then it is Fibonacci sequence?
For start I was thinking only about a sequence of form $a_{n+1} = \alpha a_n+ \beta a_{n-1}$ and all I can find is that $a_1=a_2$ and that $a_1\mid a_n$ for each $n$. Also $a_2\mid a_3$.
A sequence $(x_n)$ satisfying $\gcd(x_n,x_m)=x_{\gcd(n,m)}$ is called a strong divisibility sequence.
A simple example of a strong divisibility sequence that is not the Fibonacci sequence is $x_n = a^n-1$, where $a \in \mathbb N$.