Let a sequence be defined as follows: a_{1}=3, a_{2}=3, and for n \geq 2, a_{n+1} a_{n-1}=a_{n}^{2}+2007. Find the largest integer less than or equal to \frac{a_{2007}^{2}+a_{2006}^{2}}{a_{2007} a_{2006}}.
Let a sequence be defined as follows: a_{1}=3, a_{2}=3, and for n \geq 2, a_{n+1} a_{n-1}=a_{n}^{2}+2007. Find the largest integer less than or equal to \frac{a_{2007}^{2}+a_{2006}^{2}}{a_{2007} a_{2006}}.