IMC 2015 Problem 3

Let $F(0)=0$ , $F(1)=\frac32$ , and $F(n)=\frac{5}{2}F(n-1)-F(n-2)$ for $n\ge2$ .

Determine whether or not ${\sum_{n=0}^{\infty} \frac{1}{F(2^n)}}$ is a rational number.

(Proposed by Gerhard Woeginger, Eindhoven University of Technology)