${a}_{n+2}=2{a}_{n+1}-{a}_{n}+{2}^{n}+2,\text{}\text{}\text{}\text{}n0\text{}\text{}\text{}\text{}and\text{}\text{}\text{}{a}_{1}=1,\text{}\text{}\text{}\text{}{a}_{2}=4$

This task is in the topic of differential and difference equation. I don't know how to start solving this problem and what are we looking for? (${a}_{n},{a}_{n+2}$)

I do know how to solve the following form

${a}_{n+2}=2{a}_{n+1}-{a}_{n}$

using linear algebra as well. The actual problem I encountered the obstructionist term ${{2}^{n}+2}$

Are there some kind of variational constant method for recursive linear sequences,?

I only now this method for linear ODE with constant coefficient.

But I believe that such method could be doable here as well. Can any one provide me with a helpful hint or answer?.