Generating Function for mth Terms of Sequence/Examples/m = 3, r = 1

Example of Generating Function for mth Terms of Sequence
Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$. Let $\omega = e^{2 i \pi / 3} = -\dfrac 1 2 + \dfrac {\sqrt 3} 2 i$.

Then:
 * $a_1 z + a_4 z^4 + a_7 z^7 + \cdots = \dfrac 1 3 \left({G \left({z}\right) + \omega^{-1} G \left({\omega z}\right) + \omega^{-2} G \left({\omega^2 z}\right)}\right)$

Proof
From Generating Function for mth Terms of Sequence, for $r \in \Z$ such that $0 \le r < m$:
 * $\displaystyle \sum_{n \bmod m \mathop = r} a_n z^n = \dfrac 1 m \sum_{0 \mathop \ge k \mathop < m} \omega^{-k r} G \left({\omega^k z}\right)$

Setting $m = 3$ and $r - 1$: