What is the sum of binomial coefficients?

Question

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

@TheSilverDoe : I think the answer to "Where does your sum finish?" is clear. It continues, but only finitely many terms are non-zero. $\qquad$ — Michael Hardy, Oct 02 '18 at 15:04

score 0 · Answer 1 · answered Oct 02 '18 at 14:54

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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