Find series representation of a function $ y=\frac{1}{1+x+x^{2}+x^{3}} $ Let $y = \frac{1}{1+x+x^{2}+x^{3}}$.
And I want to find the series representation of this function.
I've noticed that I can rewrite this like $\frac{1}{1+x+x^{2}+x^{3}}=\frac{1}{1+x}*\frac{1}{1+x^{2}}$ or I can rewite this like a sum of two fractions $\frac{1}{1+x+x^{2}+x^{3}}=\frac{1}{2(1+x)}+\frac{1}{2(1+x^{2})}$.
I know the series representations of these functions $(1+x)^{n}$ and $\frac{1}{1-x}$, and I guess I should use one of them. But the result looks weird. Can you help me, please, may be I can't see some easy way to solve it.

$$f(x)=\frac{1}{1+x+x^2+x^3}=\frac{1-x}{1-x^4}=\sum_{n\geq 0}\chi(n)\,x^n $$
where $\chi(n)$ equals $1$ if $n\equiv 0\pmod{4}$, $-1$ if $n\equiv 1\pmod{4}$ and zero otherwise.
