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This question already has an answer here:

$$\binom n0 + \binom n4 + \binom n8 + \cdots$$ Any hints?

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marked as duplicate by ArsenBerk, Brahadeesh, Adrian Keister, Scientifica, Namaste Oct 2 '18 at 22:42

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ @TheSilverDoe : I think the answer to "Where does your sum finish?" is clear. It continues, but only finitely many terms are non-zero. $\qquad$ $\endgroup$ – Michael Hardy Oct 2 '18 at 15:04
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Hint: set $\omega_r=\exp(2\pi i/r)$, then look at

$$\sum_{j=0}^{r-1} (1+\omega_r^j)^n=\sum_{k=0}^n {n \choose k} \sum_{j=0}^{r-1} \omega_r^{jk}$$

That inner sum can be evaluated explicitly; it will turn out to vanish when $k$ is not a multiple of $r$.

Then set $r=4$.

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