# A complex sum!

Algebra Level 5

Let $\alpha$ be a root of the equation

$z^{23}=1$

where $\alpha\not=1$.

If the sum

$\displaystyle\sum_{k=0}^{22} \frac{1}{\alpha^{2k}+\alpha^{k}+1}$

can be expressed as $\displaystyle\frac{a}{b}$ where $a$ and $b$ are co prime positive integers find $a+b$.

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