<s>[INST] what is the value of $\binom{n}{1}​+\binom{n}{4}+\binom{n}{7}​+\binom{n}{10}+\binom{n}{13}+\dots$ what is the value of $$\binom{n}{1}​+\binom{n}{4}+\binom{n}{7}​+\binom{n}{10}+\binom{n}{13}+\dots$$ in the form of number, 
cos, sin
attempts : I can calculate the value of $$\binom{n}{0}​+\binom{n}{3}+\binom{n}{6}​+\binom{n}{9}+\binom{n}{12}+\dots=\frac{1}{3}\left(2^n+2\cos \frac{n\pi}{3}\right)$$ by use primitive $3^\text{rd}$ root of the unity
but this problem i cant solve it. 
 [/INST] As we are interested in every third term
Let $$\sum_{r=0}^{3r+1\le n}\binom n{3r+1}=\sum_{k=0}^2a_k(1+w_k)^n$$
where $w_k=w^k;k=0,1,2$ and $w$ is a complex cube root of unity so that $$1+w+w^2=0$$
Set $n=0,1,2$ to find $a_k;k=0,1,2$
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