Solve a_{2}=1/(2n+1)(2n+3)

$$a _ { 2 } = \frac { 1 } { ( 2 n + 1 ) ( 2 n + 3 ) }$$

Solve for n (complex solution)

$$n=\frac{\sqrt{a_{2}^{2}+a_{2}}}{2a_{2}}-1$$

$$n=-\frac{\sqrt{a_{2}^{2}+a_{2}}}{2a_{2}}-1\text{, }a_{2}\neq 0$$

Solve for a_2

$$a_{2}=\frac{1}{4n^{2}+8n+3}$$

$$n\neq -\frac{1}{2}\text{ and }n\neq -\frac{3}{2}$$

Solve for n

$$\left\{\begin{matrix}n=-\frac{\sqrt{a_{2}^{2}+a_{2}}}{2a_{2}}-1\text{; }n=\frac{\sqrt{a_{2}^{2}+a_{2}}}{2a_{2}}-1\text{, }&a_{2}\leq -1\\n=-\frac{\sqrt{1+\frac{1}{a_{2}}}}{2}-1\text{; }n=\frac{\sqrt{1+\frac{1}{a_{2}}}}{2}-1\text{, }&a_{2}>0\end{matrix}\right.$$