Muhammad's Recurrence

Algebra Level 5

The sequence \( \{a_n\} \) satisfies \( a_0=1, a_1=213, \) and \( a_n=2a_{n-1}+a_{n-2}\) for all \( n \geq 2 \). Let \[ S = \sum_{i=1}^{\infty} \frac{a_{i-1}}{a_i^2-a_{i-1}^2}. \] What is the value of \( \frac{1}{S} \)?

This problem is shared by Muhammad A.

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