a[0]=1; a[1]=0; a[n_]:=-2u a[n-1]-a[n-2]; Factor@Table[a[n],{n,0,20}]

a[0]=0; a[1]=1; a[n_]:=-2u a[n-1]-a[n-2]; Factor@Table[a[n],{n,0,20}]

a[0]=u; a[1]=-1; a[n_]:=-2u a[n-1]-a[n-2]; Factor@Table[a[n],{n,0,10}]

a[0]=u; a[1]=-1; a[n_]:=-2u a[n-1]-a[n-2];
Factor@PowerExpand@FullSimplify@Table[((2a[n])/(Sqrt[-((((-1+ω))^2)/ω)]))/.u->(Sqrt[(v+1)/2])/.v->-(ω+(ω^(-1)))/2,{n,0,10}]