Pattern recognition for $S_n=2+5+13+35+...+n^{th}_{term}$ I want to find sum of the first $n^{th}$ term of this sqequence .
$$2,5,13,35,97,275,793,...\\s_n=2+5+13+35+97+...$$ 
What is the closed form formula for $s_n$?

Alternatively, the sequence is:
$$2,3\cdot 2-1,3\cdot 5-2,3\cdot 13-4,3\cdot 35-8,...$$
It is the recurrence relation:
$$a_n=3a_{n-1}-2^{n-2},a_1=2.$$
Divide it by $2^{n}$:
$$\frac{a_n}{2^{n}}=\frac{3a_{n-1}}{2^{n}}-\frac14.$$
Denote: $b_n=\frac{a_n}{2^n}$ to get:
$$b_n=\frac{3}{2}b_{n-1}-\frac{1}{4},b_1=1.$$
Solution is:
$$b_n=\frac{1}{3}\left(\frac{3}{2}\right)^n+\frac12.$$
Hence:
$$a_n=2^nb_n=2^{n-1}+3^{n-1}.$$
Now the sum $S_n$ is calculated in the same way as in previous solution.
