Find the coefficient of $x^n$ simply by using binomial theorem

Question

Find the coefficient of $x^n $ in the expansion of $\frac{1}{1-x+x^2-x^3}$.

Answer 1

$\frac{1}{1-x+x^2-x^3}=\frac{1}{(1+x^2)(1-x)}=\frac{1}{2}(\frac{1+x}{1+x^2}+\frac{1} {1-x})$.

Clearly $\frac{1}{1-x}=1+x+x^2+x^3+x^4+\dots$ and

$\frac{1}{1+x^2}=\frac{1}{1-(-x^2)}=1-x^2+x^4-x^6+\dots$

from here $\frac{1+x}{1+x^2}=1+x-x^2-x^3+x^4+x^5\dots$

So $\frac{1}{1-x+x^3-x^3}=1+x+x^4+x^5+x^8+x^9+\dots$