$\displaystyle c_{0}=2,\, c_{1}=0,\,\, c_{n}=2c_{n-1}-2c_{n-2}\,\,\, n\geq2

$

$\displaystyle \sum_{n\geq2}c_{n}x^{n}=\sum_{n\geq2}2c_{n-1}x^{n}-\sum_{n\geq2}2c_{n-2}x^{n}$

$\displaystyle f(x)-2=2x(f(x)-2)-2x^{2}f(x)$

$\displaystyle f(x)-2xf(x)+2x^{2}f(x)=-4x+2$

$\displaystyle f(x)=\frac{-4x+2}{2x^{2}-2x+1}$

$\displaystyle \frac{2\pm\sqrt{4-4(2)}}{4}=\frac{2\pm2i}{4}$

I'm just not sure where to go from here. Where ever that is, I've heard you can't get there from here. Any help would be greatly appreciated.