What is the expansion for $\frac{1}{1+x+x^{2}}$?

Question

What is the expansion for $\frac{1}{1+x+x^{2}}$? I know expansion for $$\begin{align}&(1+x)^{-1}=1-x+x^{2}-x^{3}+\dots\\ &(1+x)^{-2}=1+(-2)x+(-2)\frac{-3}{2!}+(-2)(-3)\frac{-4}{3!}+\dots\\ &(1+x^{2})^{-1}=1-x^{2}+x^{4}-x^{6}+\dots\end{align}.$$ But for $\frac{1}{1+x+x^{2}}$ I got problem. Can someone derive $\frac{1}{1+x+x^{2}}$ term expansion.

Answer 1

Hint. Note that for $|x|<1$, $$\frac{1}{1+x+x^{2}}=\frac{1-x}{1-x^{3}}=(1-x)(1+x^3+x^6+x^9+\dots).$$