Question 17
Let $s_k$ denote the sum of the $k$th powers of the roots of the polynomial $x^3 - 5x^2 + 8x - 13.$ In particular, $s_0 = 3,$ $s_1 = 5,$ and $s_2 = 9.$ Let $a,$ $b,$ and $c$ be real numbers such that $s_{k+1} = a\,s_k + b\,s_{k-1} + c\,s_{k-2}$ for $k = 2,$ $3,$ $\ldots.$ What is $a + b + c?$