17. Let $S_{k}$ denote the sum of the $k$-th powers of the roots of the polynomial $x^{3}-5 x^{2}+8 x-13$, $S_{0}=3, S_{1}=5, S_{2}=9$. Let $a, b, c \in \mathbf{R}$, and
$$
S_{k+1}=a S_{k}+b S_{k-1}+c S_{k-2}(k=2,3, \cdots) \text {. }
$$

Then $a+b+c=(\quad$.
(A) -6
(B) 0
(C) 6
(D) 10
(E) 26
By Vieta's formulas, we know
$$
p+q+r=5, p q+q r+r p=8, p q r=13 \text {. }
$$

Thus, $5 S_{k}+13 S_{k-2}$
$$
\begin{aligned}
= & (p+q+r)\left(p^{k}+q^{k}+r^{k}\right)+ \\
& p q r\left(p^{k-2}+q^{k-2}+r^{k-2}\right) \\
= & p^{k+1}+q^{k+1}+r^{k+1}+p^{k} q+p^{k} r+q^{k} p+ \\
& q^{k} r+r^{k} p+r^{k} q+p^{k-1} q r+q^{k-1} p r+r^{k-1} p q \\
= & p^{k+1}+q^{k+1}+r^{k+1}+ \\
& \left(p^{k-1}+q^{k-1}+r^{k-1}\right)(p q+q r+r p) \\
= & S_{k+1}+8 S_{k-1} \\
\Rightarrow & S_{k+1}=5 S_{k}-8 S_{k-1}+13 S_{k-2} \\
& =a S_{k}+b S_{k-1}+c S_{k-2} \\
\Rightarrow & a=5, b=-8, c=13 \\
\Rightarrow & a+b+c=10 .
\end{aligned}
$$
17. D.

Let $p, q, r$ be the three roots of the polynomial $x^{3}-5 x^{2}+8 x-13$.