13. In the positive geometric sequence $\left\{a_{n}\right\}$, there exist two terms $a_{m}, a_{n}$, such that $\sqrt{a_{m} a_{n}}=8 a_{1}$, and $a_{9}=a_{8}+2 a_{7}$, then the minimum value of $\frac{1}{m}+\frac{4}{n}$ is $\qquad$.
Let the sequence $\left\{a_{n}\right\}$ have a common ratio $q, q^{2}-q-2=0 \Rightarrow q=2$. Also, $2^{\frac{m+n-2}{2}}=8 \Rightarrow m+n=8$, so when $m=3, n=5$, $\frac{1}{m}+\frac{4}{n}$ achieves its minimum value $\frac{17}{15}$.