Solving recurrence equation with generating indices of positive indices I don't know how to solve recurrence equation with positive indices like $$a_{n+2}  + 4a_{n+1}+
4a_n = 7$$ by generating functions.
How to solve such kind of problems.

To make your life easier start with the difference equation to get rid of the constant. You get (set $b_n = a_n - a_{n-1}$):
$$
b_{n+2} =-4 b_{n+1} - 4 b_n
$$
Now use GF.
EDIT
$$
a_{n+2} +4 a_{n+1} + 4 a_n = 7\\
a_{n+1} + 4 a_{n} +4 a_{n-1} = 7\\
\Delta a_{n+2} +4 \Delta a_{n+1} + 4 \Delta a_n = 0
$$
Now set $\Delta a_n = b_n$ and use GF. 
