13. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=1, a_{n}=2 a_{n-1}+n-2(n \geqslant 2) \text {. }
$$

Find the general term $a_{n}$.
Three, 13. From the given information,
$$
a_{n}+n=2\left[a_{n-1}+(n-1)\right](n \geqslant 2) \text {. }
$$

Let $b_{n}=a_{n}+n$, then
$$
b_{1}=a_{1}+1=2, b_{n}=2 b_{n-1}(n \geqslant 2) \text {. }
$$

Thus, $b_{n}=2 \times 2^{n-1}=2^{n}$, which means
$$
a_{n}+n=2^{n}(n \geqslant 2) \text {. }
$$

Therefore, $a_{n}=2^{n}-n(n \geqslant 2)$.
Since $a_{1}=1$ also satisfies the above formula, we have
$$
a_{n}=2^{n}-n(n \geqslant 1) \text {. }
$$