Form the coefficients of the inverse of a cyclotomic polynomial a periodic sequence?

Question

Interpret a cyclotomic polynomial C over Z as a formal power series over Z. C has a multiplicative inverse if and only if its constant term does not vanish.

For example the 6th cyclotomic polynomial $C_6(x) = x^2 - x + 1$ as a finite power series has the inverse power series $1 + x - x^3 - x^4 + x^6 + x^7 - \ldots$, the coefficients of which form the sequence $1, 1, 0, -1, -1, 0, 1, 1, 0, \ldots$ which is a sequence with period 6.

Many such sequences are listed in the OEIS.

Question: Constitute the coefficients of the inverse of a cyclotomic polynomial in the above sense always a periodic sequence?

Answer 1

Yes, it is periodic.

The cyclotomic polynomial $C_n(x)$ divides $x^n - 1$. If $x^n - 1 = P(x) C_n(x)$, then $$ \frac{x^n-1}{C_n(x)} = P(x)$$ If $$\frac{1}{C_n(x)} = \sum_{j=0}^\infty a_j x^j$$ that says $ a_{j+n} - a_j$ is the coefficient of $x^{n+j}$ in $P(x)$, which is $0$ since $\deg(P) < n$.