Factoring $1-(x+x²+x³+x⁴+x⁵)$

Question

I calculated the generating function $G$ of the recurrence:

$$F(0)=0$$ $$F(1)=F(2)=F(3)=F(4)=1$$ $$F(n)=F(n-1)+F(n-2)+F(n-3)+F(n-4)+F(n-5)$$

I got:

$$G(x)=\frac{x+x²+x³+x⁴}{1-(x+x²+x³+x⁴+x⁵)}$$

I want to expand this into a series to find the coefficients $\left[ X^n \right] G(X)$. But I can't find a simple way of factoring the denominator of $G(x)$. Here is what I've done so far:

$$S=1-(x+x²+x³+x⁴+x⁵)$$ $$xS=x-(x+x²+x³+x⁴+x⁵+x⁶-x)=x-(S+x⁶-x)$$ $$S(x+1)=2x-x⁶$$ $$S=\frac{2x-x⁶}{x+1}$$

This tells that $0$ is a root of $S$ but plugging $0$ back into $S$ gives $1$! I want hints.

Answer 1

You have a mistake. The correct term should be: $$S=1-(x+x^2+x^3+x^4+x^5)$$ $$xS=x-(x^2+x^3+x^4+x^5+x^6)=x-(1-S-x+x^6)=2x-x^6-1+S$$ $$S=\frac{2x-x^6-1}{x-1}$$ Therefore $x=0$ is not a root. In fact, the only real root of $S$ is: $$x\sim 0.50866$$