Complex Series!

Note that

$\begin{aligned} \sum _{ r=1 }^{ 22 } \frac { 1 }{ 1+z^{ r }+z^{ 2r } } & =\; \sum _{ r=1 }^{ 22 } \frac { z^{ r }-1 }{ z^{ 3r }-1 } \; =\; \sum _{ r=1 }^{ 22 } \frac { z^{ 24r }-1 }{ z^{ 3r }-1 } \; =\; \sum _{ r=1 }^{ 22 } \sum _{ j=0 }^{ 7 } z^{ 3jr } \\ & =\; \sum _{ j=0 }^{ 7 } \sum _{ r=1 }^{ 22 } z^{ 3jr }\; =\; 22+\sum _{ j=1 }^{ 7 } \sum _{ r=1 }^{ 22 } z^{ 3jr }\; =\; 22+\sum _{ j=1 }^{ 7 } \left( \sum _{ r=0 }^{ 22 } z^{ 3jr }-1 \right) \\ & =\; 22+\sum _{ j=1 }^{ 7 } \left( \frac { z^{ 69j }-1 }{ z^{ 3j }-1 } -1 \right) \; =\; 22-7\; =\; 15 \end{aligned}$

and hence

$\begin{aligned} \sum _{ r=0 }^{ 22 } \frac { 1 }{ 1+z^{ r }+z^{ 2r } } \end{aligned}= \frac { 1 }{ 3 } +15=\boxed{{\color{#D61F06}15\frac { 1 }{ 3 }}}$

Thanks to Mark Heninings