15. Given the sequence $\left\{a_{n}\right\}(n \geqslant 0)$ satisfies $a_{0}=0$, for all non-negative integers $n$, we have
$$
a_{n+1}=2 \sqrt{30 a_{n}\left(a_{n}+1\right)}+11 a_{n}+5 .
$$

Find the general term formula for $a_{n}$.
15. $a_{n+1}=2 \sqrt{30 a_{n}\left(a_{n}+1\right)}+11 a_{n}+5>a_{n}$.

Subtract $11 a_{n} + 5$ from both sides of $a_{n+1}-11 a_{n}-5=2 \sqrt{30 a_{n}\left(a_{n}+1\right)}$ and square both sides, then rearrange to get
$$
\begin{array}{l}
a_{n}^{2}+a_{n+1}^{2}-22 a_{n} a_{n+1}-10 a_{n}-10 a_{n+1}+25=0 . \\
\text { Also, } a_{n-1}^{2}+a_{n}^{2}-22 a_{n-1} a_{n}-10 a_{n-1}-10 a_{n}+25=0,
\end{array}
$$

Subtract the two equations to get
$$
\begin{array}{l}
a_{n+1}^{2}-a_{n-1}^{2}-22 a_{n}\left(a_{n+1}-a_{n-1}\right)-10\left(a_{n+1}-a_{n-1}\right) \\
=0 .
\end{array}
$$

Since $a_{n+1}-a_{n-1}>0$, we have
$$
a_{n+1}+a_{n-1}-22 a_{n}-10=0 \text {. }
$$

Because $a_{n}+a_{n-2}-22 a_{n-1}-10=0$, subtract the two equations to get the homogeneous linear recurrence relation
$$
a_{n+1}-23 a_{n}+23 a_{n-1}-a_{n-2}=0 .
$$

The characteristic equation is $x^{3}-23 x^{2}+23 x-1=0$, with characteristic roots
$$
\lambda_{1}=1, \lambda_{2}=11+2 \sqrt{30}, \lambda_{3}=11-2 \sqrt{30} \text {. }
$$

Assume $a_{n}=C_{1}+C_{2}(11+2 \sqrt{30})^{n}+C_{3}(11-2 \sqrt{30})^{n}$.
Given $a_{0}=0, a_{1}=5, a_{2}=120$, we get
$$
C_{1}=-\frac{1}{2}, C_{2}=C_{3}=\frac{1}{4} \text {. }
$$

Therefore, the general term formula is
$$
a_{n}=-\frac{1}{2}+\frac{1}{4}(11+2 \sqrt{30})^{n}+\frac{1}{4}(11-2 \sqrt{30})^{n} \text {. }
$$
(Provided by An Hai)