Matt's Recurrence

Geometry Level 5

Let θ=sin1725 \theta = \sin ^{-1} \frac {7}{25} . Consider the sequence of values defined by an=sin(nθ) a_n = \sin ( n \theta) . They satisfy the recurrence relation an+2=k1an+1+k0an,nN a_{n+2} = k_1 a_{n+1} + k_0 a_{ n}, \quad n \in \mathbb{N} for some (fixed) real numbers k1,k0 k_1 , k_0 . The sum k1+k0 k_1 + k_0 can be written as pq \frac {p}{q} , where p p and qq are positive coprime integers. What is the value of p+q p + q ?

This problem is proposed by Matt.

Details and assumptions

By definition, sin1:[1,1][π2,π2] \sin ^{-1} : [-1 ,1 ] \rightarrow [ - \frac {\pi}{2} , \frac {\pi}{2} ] .

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