Generalized Fibonacci number

Hint: If $f(x) = \sum_{n=0}^\infty a_n x^n$, find a relation between $f(x)$, $x f(x)$ and $x^2 f(x)$.

Expanding on the hint provided in the answer by Robert Israel, note it is much easier to compare the values of $f(x)$, $xf(x)$ and $x^2 f(x)$ after you first adjust them so they each use an infinite series with the same exponents for $x$. In particular, you can rewrite them as follows:

$$f(x) = a_0 + a_1 x + \sum_{n=0}^{\infty}a_{n+2} x^{n+2} \tag{1}\label{eq1}$$ $$xf(x) = a_0 x + \sum_{n=0}^{\infty}a_{n+1} x^{n+2} \tag{2}\label{eq2}$$ $$x^2 f(x) = \sum_{n=0}^{\infty}a_{n} x^{n+2} \tag{3}\label{eq3}$$

With these equations, you can use the recursive definition for $a_{n+2}$ to determine an equation from which you can get $f(x)$ in closed form.