Recurring Styles #12 - Limit of a Recurrence!

Calculus Level 5

\[\large{ S = \sum_{n=1}^\infty \dfrac{a_{2n+2}}{a^2_{n-1} a^2_{n+1}} }\]

Let \((a_n)\) be a sequence defined by \(a_0 = 1, a_1 = 2\), and for \(n \geq 2\), \(a_n = a_{n-1} + a_{n-2}\). If \(S\) can be expressed as \(\dfrac{A}{B} \) where \(A,B \in \mathbb Z^+, \ \gcd(A,B)=1\). Submit the value of \(A+B\) as your answer.

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