Question 176
Let $f_n(x)$ be a sequence of polynomials, where $f_0(x) = 2$, $f_1(x) = 3x$, and
$$
f_n(x) = 3x f_{n-1}(x) + (1 - x - 2x^2) f_{n-2}(x)
$$
for all $n \ge 2$. Determine all positive integers $n$ such that $f_n(x)$ is divisible by $x^3 - x^2 + x$.
$$
f_n(x) = 3x f_{n-1}(x) + (1 - x - 2x^2) f_{n-2}(x)
$$
for all $n \ge 2$. Determine all positive integers $n$ such that $f_n(x)$ is divisible by $x^3 - x^2 + x$.