$$
\begin{array}{r}
\text { 1. If } 2008=a_{n}(-3)^{n}+a_{n-1}(-3)^{n-1}+ \\
\cdots+a_{1}(-3)+a_{0}\left(a_{i}=0, \pm 1, \pm 2, i=0,1\right. \text {, }
\end{array}
$$
$\cdots, n)$, then $a_{n}+a_{n-1}+\cdots+a_{1}+a_{0}=$ $\qquad$
1.0 or $\pm 4$ or $\pm 8$.
$$
\begin{aligned}
2008= & 2(-3)^{6}-2(-3)^{5}-2(-3)^{3}+ \\
& (-3)^{2}+1,
\end{aligned}
$$

At this point, $a_{n}+a_{n-1}+\cdots+a_{0}=0$;
$$
\begin{aligned}
2008= & 2(-3)^{6}-2(-3)^{5}-2(-3)^{3}+ \\
& (-3)^{2}-(-3)-2,
\end{aligned}
$$

At this point, $a_{n}+a_{n-1}+\cdots+a_{0}=-4$;
$$
\begin{aligned}
2008= & -(-3)^{7}-(-3)^{6}-2(-3)^{5}- \\
& 2(-3)^{3}+(-3)^{2}-(-3)-2,
\end{aligned}
$$

At this point, $a_{n}+a_{n-1}+\cdots+a_{0}=-8$;
$$
\begin{aligned}
2008= & 2(-3)^{6}-2(-3)^{5}+(-3)^{4}+ \\
& (-3)^{3}+(-3)^{2}+1,
\end{aligned}
$$

At this point, $a_{n}+a_{n-1}+\cdots+a_{0}=4$;
$$
\begin{aligned}
2008= & (-3)^{8}+2(-3)^{7}+(-3)^{5}+ \\
& (-3)^{4}+(-3)^{3}+(-3)^{2}+1,
\end{aligned}
$$

At this point, $a_{n}+a_{n-1}+\cdots+a_{0}=8$.
Notice that
when $(-3)^{n}$ is changed to $(-1)(-3)^{n+1}-2(-3)^{n}$,
when $2(-3)^{n}$ is changed to $(-1)(-3)^{n+1}-(-3)^{n}$,
when $3(-3)^{n}$ is changed to $(-1)(-3)^{n+1}$,

the value of $a_{n}+a_{n-1}+\cdots+a_{0}$ increases or decreases by 4, and when $n>8$, the absolute value of $a_{n}+a_{n-1}+\cdots+a_{0}$ is no greater than 8. Therefore, $a_{n}+a_{n-1}+\cdots+a_{0}=$ 0 or $\pm 4$ or $\pm 8$.