Fibonacci fun

$F_n = \sqrt { { F }_{ 2001 }{ F }_{ 1999 }+\frac { { F }_{ 2016 }{ F }_{ 2000 }-{ F }_{ 2010 }{ F }_{ 2006 } }{ 440 } }$

Let $F_n$ denote the $n^\text{th}$ Fibonacci number . Find the value of $n$ such that the equation above is fulfilled.

This problem is original.

Picture credits: Aloe polyphylla spiral by Just chaos, Wikipedia