Faraz's reciprocal sequence

Algebra Level 5

Consider the sequence {ai}i0\{a_i\}_{i\geq0} such that a0=7+1a_0=\sqrt{7}+1 and an+1=an2an+1a_{n+1}=a_n^2-a_n+1. The sum n=01an,\sum_{n=0}^{\infty} \frac{1}{a_n}, can be written as 1b\frac{1}{\sqrt{b}}. What is the value of bb?

This problem is shared by Faraz M. from AMSP.

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