I want to range transform $2^n=na_n+na_{n-1}-a_{n-1}$ to get rid of the $2^n$ term and then solve it with any other method (seems like telescoping will work once it's reduced).
I've tried transforming it by $b_n=a_n/2^n$ which yields:
$1=\frac{n}{2^n}*b_n+\frac{n}{2^n}*b_{n-1}-b_{n-1}$
But now the $b_n$ terms have $2^n$ in their coefficient, which is no better than I started with.
Hint First simplify the equation, you get $2^n=na_n+(n-1)a_{n-1}$ and let $u_n=(-1)^nna_n$ then $2^n(-1)^n=u_n-u_{n-1}$ so that you have: $$u_{n}=\sum_{i=1}^n(-1)^i2^i+u_0=\frac{2}{3}\left((-2)^n-1\right)$$
so that $$a_n=\frac{2}{3n}\left((2)^n-(-1)^n\right)$$