Solving this recurrence relation

Question

please tell me a way to solve this recurrence $$n\cdot R_n=C_1\cdot R_{n-1}+C_2\cdot R_{n-2}.$$ $C_1$ and $C_2$ are constants.. There is an $n$ there in the left hand side.. it makes mess. I tried generating functions too.. But the issue is the last $n\cdot R_n$ can be written as $\frac{d}{dz}$ of $A(z)$ (generating function).

Answer 1

Hint: $$\sum^{\infty}_{n=0}(n+2)R_{n+2}x^n=\frac{1}{x^2}\left(\sum^{\infty}_{n=0}nR_nx^n-R_1x\right)=\frac{xR'(x)-R_1x}{x^2}$$ where $$R(x)=\sum^{\infty}_{n=0}R_nx^n$$