Does this really have a closed form?

Algebra Level 4

\[ \large 22 \sum_{n=0}^\infty \dfrac{F_n}{4^n} \] Let \(F_n\) denote the \(n^\text{th} \) Fibonacci number, where \(F_0 = 0, F_1 = 1\) and \(F_n = F_{n-1} + F_{n-2} \) for \(n=2,3,4,\ldots \).

Compute the expression above.

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