What gon??!!

There is a regular $n$ -sided polygon $A_0 A_1 A_2 ... A_{n-1}$ for which the following equation holds:

$\frac{1}{A_{0}A_{1}}=\frac{1}{A_{0}A_{2}}+\frac{1}{A_{0}A_{4}}\ +\ \frac{1}{A_{0}A_{8}}+\frac{1}{A_{0}A_{15}}$

Find the value of $n$ given that it is greater than $16$ .

Note :
The notation $A_{i}A_{j}$ indicates the distance between the $i^{th}$ and the $j^{th}$ vertices.