More fun in 2016, Part 1

Algebra Level 4

$S=\sum_{k=0}^{672}{2016 \choose 3k}$ The sum $$S$$ is of the form $$S=\dfrac{2^{2016}+a}{3}$$. Find $$a$$.

Bonus Question: What is $S=\sum_{k=0}^{\lfloor{n/3}\rfloor}{n \choose 3k}$ for any positive integer $$n$$?

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