If $f(x+1)+f(x-1)=\sqrt 3 f(x), \forall x$ then $f$ is periodic. 
If $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
  $f(x+1)+f(x-1)=\sqrt 3 f(x), \forall x$ then $f$ is periodic.


I tried to replace $x$ by $x+1, x-1$ in the equality,to get something like $f(x + k)=f(x)$ but without success.
Any help is appreciated.

Fix some $x \in \Bbb R$.
Define $a_n = f(x+n)$, for $n \in \Bbb N$.
Note that $a_n$ satisfies the recurrence relation $a_{n+1} = \sqrt{3} a_n - a_{n-1}$. By the general theory of recurrence relations it follows that $a_n=c_1 r_1^n + c_2 r_2^n$ where $r_1,r_2$ are the roots of the quadratic equation $r^2-\sqrt{3}r+1=0$.
But note that $r_1,r_2$ are complex numbers with norm 1 (specifically, $\pm e^{i\pi/6}$). Hence it follows that $a_n = A\cos (\pi n/6) + B \sin(\pi n /6)$. Thus you can conclude that $a_{n+12}=a_n$ for all $n$, i.e, $f(x+12)=f(x)$.
