Cardinality of a set with a recurrence relation. Let $A = \left\{ f\in \mathbb{N}\rightarrow \mathbb{C} \mid \forall n\in \mathbb{N}. f(n+3) + 3f(n+1) = f(n+2)+f(n) \right\}$
What is $\left|A\right|$?  
Well, I tried to treat $f$ as a recurrence relation and hence, the characteristic polynomial is: $x^3 + 3x = x^2 + 1$.  
The roots for this polynomial aren't "nice". 
So, I guess I should do something different, though it seems very natural to use characteristic polynomial here.

This recurrence defines $f(n)$ for all $n \in \mathbb{N}$ given $f(0), f(1), f(2)$. For each set of those it gives a different function, so...
