I want to solve a generating function - recursive sequence problem and I am stuck one step before the solution where I want to combine the three sums in order to extract a formula for $a_n$.
$$G(x) = 4\Sigma_{n=0}^\infty 3^nx^n - 4\Sigma_{n=1}^\infty 3^nx^n + \Sigma_{n=2}^\infty 3^nx^n$$
A hint would be great since I know I am close but I can't see the way to get from the above to a format like
$$G(x) = \Sigma_{n=0}^\infty a(c+...+d) 3^nx^n $$
Let $(a_0,a_1,\dots)$ denote $\sum_{n=0}^\infty a_n3^nx^n$. There are three terms in $G(x)$; from left to right they are $$(4,4,4,4,4,\dots)$$ $$(0,-4,-4,-4,-4,\dots)$$ $$(0,0,1,1,1,\dots)$$ They sum to the following sequence: $$(4,0,1,1,1,\dots)$$ i.e. $$G(x)=4+\sum_{n=2}^\infty3^nx^n=4+\frac{(3x)^2}{1-3x}$$