Consider the sequence a_1, a_2, a_3, … such that a_1=1, a_2=2 and a_n+2=2/(a_n+1)+a_n for n=1,2,3, …. If ((a_1+1/a_2)/a_3) ·((a_2+1/a_3)/a_4)…

Consider the sequence $\displaystyle a_1, a_2, a_3, \ldots$ such that $\displaystyle a_1=1, a_2=2$ and $\displaystyle a_{n+2}=\frac{2}{a_{n+1}}+a_n$ for $\displaystyle n=1,2,3, \ldots$. If $\displaystyle \left(\frac{a_1+\dfrac{1}{a_2}}{a_3}\right) \cdot\left(\frac{a_2+\dfrac{1}{a_3}}{a_4}\right) \cdot\left(\frac{a_3+\dfrac{1}{a_4}}{a_5}\right) \ldots .\left(\frac{a_{30}+\dfrac{1}{a_{31}}}{a_{32}}\right)=2^\alpha\left({ }^{61} C_{31}\right)$, then $\displaystyle \alpha$ is equal to :