CN-TST 2018 Problem 19

Define the polymonial sequence $\left \{ f_n\left ( x \right ) \right \}_{n\ge 1}$ with $f_1\left ( x \right )=1$ , $$f_{2n}\left ( x \right )=xf_n\left ( x \right ), f_{2n+1}\left ( x \right ) = f_n\left ( x \right )+ f_{n+1} \left ( x \right ), n\ge 1.$$ Look for all the rational number $a$ which is a root of certain $f_n\left ( x \right ).$