Matt's Recurrence

Geometry Level 5

Let \( \theta = \sin ^{-1} \frac {7}{25} \). Consider the sequence of values defined by \( a_n = \sin ( n \theta) \). They satisfy the recurrence relation \[ a_{n+2} = k_1 a_{n+1} + k_0 a_{ n}, \quad n \in \mathbb{N} \] for some (fixed) real numbers \( k_1 , k_0 \). The sum \( k_1 + k_0 \) can be written as \( \frac {p}{q} \), where \( p\) and \(q\) are positive coprime integers. What is the value of \( p + q \)?

This problem is proposed by Matt.

Details and assumptions

By definition, \( \sin ^{-1} : [-1 ,1 ] \rightarrow [ - \frac {\pi}{2} , \frac {\pi}{2} ] \).

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