Let $x_{0}$, $x_{1}$, $x_{2}$, $...$ , $x_{2018}$ be the sequence of positive integers satisfying the three conditions below:
I) $\ 1 = x_{0} \le x_{1} \le x_{2} \le \ ... \le x_{2018}$
II) $\ $ The range of the sequence consists of exactly 100 different positive integers
III) $\ \sum_{i=2}^{2018} x_i (x_i - x_{i-2}) = 9998$
What is the number of distinct sequences $x_{0}$, $x_{1}$, $x_{2}$, $...$ , $x_{2018}$?
I do not know how to approach this recreational math problem even though I spent quite some time on it. All I have drawn upon given information is the monotonicity of the sequence with $(1)$ and this oddly, though it may point to a hint, interesting relation between $100$ and $9998$ in $100^{2} - 2 = 9998$.
How can I move on from here?