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.

There are two possibilities here: either we use the binomial theorem on $(1+x(1+x+x^2))^{-1}$, which looks awful, or we use a trick: multiplying and dividing by $1-x$, we have
$$ \frac{1-x}{(1-x)(1+x+x^2+x^3)} = \frac{1-x}{1-x^4}, $$
because the middle terms all cancel out. Now, we can expand the denominator,
$$ \frac{1-x}{1-x^4} = (1-x)\sum_{k=0}^{\infty} x^{4k}, $$
and it's pretty easy to multiply through and get the final answer from here.
