Faraz's reciprocal sequence

Algebra Level 5

Consider the sequence \(\{a_i\}_{i\geq0}\) such that \(a_0=\sqrt{7}+1\) and \(a_{n+1}=a_n^2-a_n+1\). The sum \[\sum_{i=0}^{\infty} \frac{1}{a_n},\] can be written as \(\frac{1}{\sqrt{b}}\). What is the value of \(b\)?

This problem is shared by Faraz M. from AMSP.

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