This series satisfies the following functional equatioin
$$(g(x))^{2} =2\sum_{r=1}^{\infty}g(x^{3^{r}+1})+g(x^{2})$$
We are working on a algebraic problem we have finally ended up with this series $g(x)$. We have not been successful in finding the closed form so far.
I have attached some of our work to find the closed form for the series $g(x)$
