Solving linear inhomogeneous recurrence with 3^n term

[MarkFL]

I have again merged threads...please post questions pertaining to this problem in this thread. :D

 

[Evgeny.Makarov]

Unfortunately, I am also not very good with generating functions and have little time. I can only recommend the following formulas.
\begin{align}
(x^{n+1})'&=nx^n+x^n&&\text{to find }\sum(-1)^nnx^n\\
(x^{n+2})''&=n^2x^n+3nx^n+2x^n&&\text{to find }\sum(-1)^nn^2x^n\\
\sum n(3x)^n&=\sum n(3^nx^n)
\end{align}

 

[johng1]

The calculation of a closed form for the generating function G(z) is straight forward. This is a rational function in z. If you're really ambitious (I'm not), you can apply partial fraction decomposition to G(z), use the geometric series and it's derivatives and find the closed form solution for $a_n$.