Let $f(x)=(x^2-x-1)^n=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0.$ Now, let $y=a_1+a_3+...+a_{2n-1}$ and $x=a_0+a_2+a_4+...+a_{2n}.$ The key to this problem is to realize that $f(1)=x+y$ and $f(-1)=x-y.$ Thus, we find that $f(1)=(-1)^n=x+y$ and $f(-1)=(1)^n=x-y.$ Finally, we conclude that $x=\frac{(-1)^n+1}{2}$ and $y=\frac{(-1)^n-1}{2}.$