8. Given a sequence of non-negative integers $\left\{a_{n}\right\}$ satisfying
$$
a_{1}=2016, a_{n+1} \leqslant \sqrt{a_{n}},
$$

and if the number of terms is no less than 2, then any two terms are distinct. Then, the number of such sequences $\left\{a_{n}\right\}$ is $\qquad$
8. 948 .

Let the number of non-negative integer sequences $\left\{a_{n}\right\}$ satisfying $a_{1}=m, a_{n+1} \leqslant \sqrt{a_{n}}$ be $b_{m}$. Then
$$
b_{2016}=b_{44}+b_{43}+\cdots+b_{0}+1 \text {. }
$$

Through calculation, we get
$$
\begin{array}{l}
b_{0}=1, b_{1}=2, \\
b_{2}=b_{3}=b_{1}+b_{0}+1=4, \\
b_{4}=b_{5}=\cdots=b_{8}=b_{2}+b_{1}+b_{0}+1=8, \\
b_{9}=b_{10}=\cdots=b_{15}=b_{3}+b_{2}+b_{1}+b_{0}+1=12, \\
b_{16}=b_{17}=\cdots=b_{24}=b_{4}+b_{3}+\cdots+b_{0}+1=20, \\
b_{25}=b_{26}=\cdots=b_{35}=b_{5}+b_{4}+\cdots+b_{0}+1=28, \\
b_{36}=b_{37}=\cdots=b_{44}=b_{6}+b_{5}+\cdots+b_{0}+1=36 .
\end{array}
$$

Therefore, $b_{2016}=b_{44}+b_{43}+\cdots+b_{0}+1$
$$
\begin{array}{c}
=36 \times 9+28 \times 11+20 \times 9+12 \times 7+ \\
8 \times 5+4 \times 2+2+1+1=948 .
\end{array}
$$

Thus, the number of such sequences $\left\{a_{n}\right\}$ is 948 .