Let $\alpha$ denote $\cos^{-1}(\tfrac{2}{3})$. The recursive sequence $a_0, a_1, a_2, \ldots$ satisfies $a_0 = 1$ and, for all positive integers $n$,  
$$a_n = \dfrac{\cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1)}{2a_0}.$$  
Suppose that the series  
$$\sum_{k=0}^\infty\dfrac{a_k}{2^k}$$  
can be expressed uniquely as $\tfrac{p\sqrt{q}}{r}$, where $p$ and $r$ are coprime positive integers and $q$ is not divisible by the square of any prime. Find the value of $p+q+r$.

Let's think step by step and output the final answer within \boxed{}.
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