Height of Elegance!

The sequence of integers ${ \left\{ { a }_{ i } \right\} }_{ i=0 }^{ \infty }$ satisfies $a_0 = 3$ , $a_1 = 4$ , and

$\large\ { a }_{ n+2 } = { a }_{ n+1 }{ a }_{ n } + \left\lceil \sqrt { { a }_{ n+1 }^{ 2 } - 1 } \sqrt { { a }_{ n }^{ 2 } - 1 } \right\rceil$

for $n \ge 0$ .

Evaluate the sum :

$\large\ \displaystyle \sum _{ n=0 }^{ \infty }{ \left( \frac { { a }_{ n+3 } }{ { a }_{ n+2 } } - \frac { { a }_{ n+2 } }{ { a }_{ n } } + \frac { { a }_{ n+1 } }{ { a }_{ n+3 } } - \frac { { a }_{ n } }{ { a }_{ n+1 } } \right) } .$