Floor Functions in Recursion.

${ a }_{ n } = \left\lfloor { \left( 2018 + \sqrt { { 2018 }^{ 2 } + 1 } \right) }^{ n } + { \left( \frac { 1 }{ 2 } \right) }^{ n } \right\rfloor$

Define $a_n$ as above for all non-negative integers $n$ . Find $2018^2 \displaystyle \sum _{ n=1 }^{ \infty }{ \frac { 1 }{ { a }_{ n - 1 }{ a }_{ n + 1 } } }$ .

Notation: $\lfloor \cdot \rfloor$ denotes the floor function .