partial Fibonacci summation Let $F_{n}$ be the n-th Fibonacci number.
How to calculate the summation like following:
$\sum_{n \geq 0}  F_{3n} \cdot 2^{-3n}$

$F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_n + F_{n+1}$. 
$F_{n+3} = F_n + 2F_{n+1}$
$F_{n+6} = 5F_n + 8F_{n+1} = F_n + 4F_{n+3}$
Let G be a series where $G_0 = 0$, $G_1 = 2$, and $G_{n+2} = G_n + 4G_{n+1}$.  In other words, $G_i = F_{3i}$.  Its generating function is $g(x) = \dfrac{2x}{1 - 4x - x^2}$.
$\sum_{n \ge 0}F_{3n}2^{-3n} = \sum_{n \ge 0} G_n{\left(\dfrac18\right)}^{n} = g(\dfrac18) = \dfrac{16}{31}$
