# Messy sum simplifies to a binomial coefficient!

Let $S(n)=\sum_{i=0}^n 2^i{n\choose i}{n-i\choose \left\lfloor\frac{n-i}{2}\right\rfloor}.$

If $$S(2017) = \dbinom{M}{2017}$$, find $$M$$.


Notations:

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