> 
> Can we expect to find no-zero function $ \ f:\mathbb Z \to \mathbb C$ such that $f(n)= -f(-n+m)$ for $m\in \mathbb Z \setminus 3\mathbb Z$ and $n\in \mathbb Z$?
> 
> 
> 

**Edit**: the example in my mind is: $f(n)=(-1)^n= (e^{\pi i})^n$, this satisfy the above situation if we take $m\in \mathbb Z \setminus 2\mathbb Z.$ I'm willing to generalize the property of $f$ for $m\in \mathbb Z \setminus 3\mathbb Z.$