While doing a combinatorial problem, with $n$ being even, I came up with the expression

$$\sum\_{i=0}^{n/2} {n-i\choose i}2^i$$

for which I used wolfram to get a closed form expression of $\dfrac{1}{3}\left(2^{n+1}+(-1)^n\right)$. 

Is there an easy way to obtain this closed-form?

Also, if there are any good references for binomial coefficient identities like these I'd appreciate it. I searched some but did not find any similar to this.