Summation+Nested radical 2

$f(x)=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}}$

Let $f(x)=a$ where $a$ is a positive integer.

$\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}}=a$

$\Rightarrow \sqrt{x+a}=a$

$\Rightarrow x+a=a^{2}$

$\Rightarrow a^{2}-a-x=0$

$\Rightarrow a=\frac{1+\sqrt{1+4x}}{2}$

If we want $a$ to be an integer, $\sqrt{1+4x}$ should be a perfect square. Therefore, values of $x$ are as follows :

$x=\underbrace{0,2,6,12,20,30}_{\text{6 values}}$

$\Rightarrow \displaystyle \sum^{216}_{n=1}\frac{n(n+1)}{2}=\frac{1}{2} \displaystyle \sum^{216}_{n=1}n^2+n$

$\Rightarrow \frac{1}{2}\displaystyle \sum^{216}_{n=1}n^2+\frac{1}{2}\displaystyle \sum^{216}_{n=1}n$

$\Rightarrow \frac{1}{2}\cdot \frac{216×217×433}{6}+\frac{1}{2}\cdot \frac{216×217}{2}$

$\Rightarrow \boxed{1703016}$