How find this $\sum_{k=0}^{100}a_{3k}$

Question

let

$$(1+x+x^2)^{150}=\sum_{k=0}^{300}a_{k}x^k$$

find the value

$$\sum_{k=0}^{100}a_{3k}$$

My idea: I think

we let $x=1,w,w^2$?

Answer 1

Yes. $\frac13(f(1)+f(w)+f(w^2))=3^{149}$ is the desired result where $w$ is a primitive third root of unity.