Recursion in between

$\large{{ a }_{ n+1 }=\left\lfloor \frac { 4 }{ 3 } \left( \left\lfloor \sqrt { { a }_{ n }^{ 2 }+5{ a }_{ n } } \right\rfloor \right) \right\rfloor }$

Let there be a sequence defined by above rule for $n\ge 1$ . If $a_{16}=628$ and the first perfect square immediately before $a_{16}$ occurs at $a_{x}$ ( $x<14$ ) and $\large{\left\lfloor \frac { { a }_{ 20 } }{ { a }_{ 10 } } \right\rfloor =y}$ . Find $x+y$