Generating Function for mth Terms of Sequence/Examples

Examples of Generating Function for mth Terms of Sequence

Example: $m = 3, r = 1$

Let $\omega = e^{2 i \pi / 3} = -\dfrac 1 2 + \dfrac {\sqrt 3} 2 i$.

Then:

$a_1 z + a_4 z^4 + a_7 z^7 + \cdots = \dfrac 1 3 \paren {\map G z + \omega^{-1} \map G {\omega z} + \omega^{-2} \map G {\omega^2 z} }$