Muhammad's Recurrence

Algebra Level 5

The sequence {an} \{a_n\} satisfies a0=1,a1=213, a_0=1, a_1=213, and an=2an1+an2 a_n=2a_{n-1}+a_{n-2} for all n2 n \geq 2 . Let S=i=1ai1ai2ai12. S = \sum_{i=1}^{\infty} \frac{a_{i-1}}{a_i^2-a_{i-1}^2}. What is the value of 1S \frac{1}{S} ?

This problem is shared by Muhammad A.

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