I'm trying to solve this linear recurrence with generating functions, but I keep getting stuck on the last few steps. I found the generating function, but after splitting it into partial fractions and putting it in sigma notation, I don't know how to simplify it into one summation:
$g_0$=1, $g_1$=2, $g_2$=-2
$g_n$=7$g_{n-1}$-16$g_{n-2}$+12$g_{n-3}$
I got up to the point;
$$\frac{1-5z}{1-7z+16z^2-12z^3}=\frac{7-8z}{(1-2z)^2}+\frac{-6}{1-3z}$$
and finally;
$$=(7-8z)\sum_{n=0}^\infty(n+1)2^nz^n-6\sum_{n=0}^\infty3^nz^n$$
I have no clue what to do now, but the final result should be;
$$g_n=(3n+7)\cdot2^n-6\cdot3^n$$ The question is from here if I missed anything (pg 15-18). Any help would be GREATLY aprreciated.
$$\begin{align*} (7-8z)\sum_{n\ge 0}(n+1)2^nz^n&=7\sum_{n\ge 0}(n+1)2^nz^n-8\sum_{n\ge 0}(n+1)2^nz^{n+1}\\\\ &=7\sum_{n\ge 0}(n+1)2^nz^n-4\sum_{n\ge 0}(n+1)2^{n+1}z^{n+1}\\\\ &=7\sum_{n\ge 0}(n+1)2^nz^n-4\sum_{n\ge 1}n2^nz^n\\\\ &=7+\sum_{n\ge 1}7(n+1)2^nz^n-\sum_{n\ge 1}4n2^nz^n\\\\ &=7+\sum_{n\ge 1}(3n+7)2^nz^n\\\\ &=\sum_{n\ge 0}(3n+7)2^nz^n\;, \end{align*}$$
so
$$(7-8z)\sum_{n\ge 0}(n+1)2^nz^n-6\sum_{n\ge 0}3^nz^n=\sum_{n\ge 0}\Big((3n+7)2^n-6\cdot3^n\Big)z^n\;,$$
and $g_n=(3n+7)2^n-6\cdot3^n$.