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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