16. Solve the following recurrence relations:
(1) $\left\{\begin{array}{l}a_{n}=2 a_{n-1}+2^{n} \quad(n \geqslant 1), \\ a_{0}=3 .\end{array}\right.$
(2) $\left\{\begin{array}{l}a_{n}=n a_{n-1}+(-1)^{n} \quad(n \geqslant 1), \\ a_{0}=3 .\end{array}\right.$
(3) $\left\{\begin{array}{l}a_{n}=2 a_{n-1}-1 \quad(n \geqslant 1), \\ a_{0}=2 .\end{array}\right.$
(4) $\left\{\begin{array}{l}a_{n}=\frac{1}{2} a_{n-1}+\frac{1}{2^{n}} \quad(n \geqslant 1), \\ a_{0}=1 .\end{array}\right.$
(5) $\left\{\begin{array}{l}a_{n}=4 a_{n-2} \quad(n \geqslant 2), \\ a_{0}=2, a_{1}=1 .\end{array}\right.$
16. (1) $a_{n}=(n+3) 2^{n}$.
(2) $a_{n}=n!\left(2+\sum_{k=0}^{n}(-1)^{k} / k!\right)$.
(3) $a_{n}=2^{n}+1$.
(4) $a_{n}=\frac{n+1}{2^{n}}$.
(5) $a_{n}=2^{n+(-1)^{n}}$.