ILL 1990 Problem 22

Let $f(0) = f(1) = 0$ and

$$f(n+2) = 4^{n+2} \cdot f(n+1) - 16^{n+1} \cdot f(n) + n \cdot 2^{n^2}, n = 0, 1, 2, \ldots$$

Show that the numbers $f(1989), f(1990), f(1991)$ are divisible by $13.$