Question 57
Let $n \ge 2$ be an integer and $z$ be a complex number so that $z^n = 1$. Prove that:
$$ \prod_{k=1}^{n} (1 - z^k - z^{2k}) = \left\lfloor \left( \frac{1 + \sqrt{5}}{2} \right)^n \right\rfloor - \frac{1 + (-1)^n}{2} $$
$$ \prod_{k=1}^{n} (1 - z^k - z^{2k}) = \left\lfloor \left( \frac{1 + \sqrt{5}}{2} \right)^n \right\rfloor - \frac{1 + (-1)^n}{2} $$