Note that: $$a^2+\left ( \frac{a^2}{4}-1 \right )^2=\left ( \frac{a^2}{4}+1 \right )^2; a^2 +\left ( \frac{a^2-1}{2} \right )^2=\left ( \frac{a^2+1}{2} \right )^2.$$Consider $(x_n):$ $x_1=c, \,x_{n+1}=\left[\begin{matrix} & \frac{x_n^2-1}{2} \textrm{ if } x_n \textrm{ is odd.} & \\& \frac{x_n^2}{4}-1 \textrm{ if } x_n \textrm{ is even.} \end{matrix}\right.$ Look at $n_k=x_k^2+x_{k+1}^2+...+x_{k+2017}^2$ and note that $x_i^2 +x_{i+1}^2 $ is a perfect square for all $i=\overline{k,k+2016.}$ So $n_k$ can be represented in the form sum of $2017$ perfect squares, at least $2017$ dinstinct ways, which is needed.