Also, is it ok if I add a note at the end of the original question to add in this condition, or should I just start a new post? I'm not sure what MO etiquette is.

Thanks very much for your extremely detailed response! Can I please be annoying and add the condition that $a_1, ... a_n$ is a sequence of non-negative integers? The argument I had in mind in the cyclic case was to construct a polynomial $f(x)$ with coefficients $a_i$ like the one you did ($g^2 - g + 1$), and you'd then have that $f(x)$ divides $g(x) = \frac{x^n-1}{x-1}$. However, as $g(1) = n$ and $f(1)$ is relatively prime to $n$, this is a contradiction -- unless of course $f(1) = 1$ -- and the condition $a_i \geq 0$ deals with that case.