Let's consider a sum $$S_{m}=\sum_{ 3 | n}^{m} {a_{n}}$$ where the sum is taken over all the integers $3t$, where $0 \leq 3t \leq m$. Assume that $G(z)$ is a generating function of the sequence $b_{m}=\sum_{n=0}^{m}{a_{n}}$. How to derive a generating function $$G_{1}(z)=\sum_{k=0}^{\infty}{S_{k} z^{k}}$$ with the knowledge of $G(z)$?
For the sum $S_{m}= \sum_{2 | n}^{m}{a_{n}}$ and corresponding $A(z)=\sum_{t=0}^{\infty}{\sum_{n}^{t}{a_{n}} \cdot z^{t}}$ it can be done: $G(z)=\frac{1}{2}(A(\sqrt{z})+A(-\sqrt{z}))$.
Any help would be much appreciated.