Trying to solve the following recurrence relation.

Question

I have been trying to solve the following recurrence relation: $a_{n+2}=4a_{n+1}-4a_n+n^2-5n+2$. I already figured out that the general solution should be $c_12^n+c_2n2^n$ because the root 2 has multiplicity 2. But I'm having hard time finding the particular solution to this recurrence relation. Any help will be appreciated.

Answer 1

To get rid of the value of $y_0$, define $u_n=\frac {y_n}{y_0}$ and later $v_n=u_n^2$ to face $$v_{n+1}=\frac{n+3}{n+1}\,v_n\qquad \text{with} \qquad v_0=1$$ Comupute the very first terms and notice that they are along a quadratic in $n$. So, let $v_n=a+b n+c n^2$ and identify the coefficients.