The problem is to calculate the sum \[\binom{n}{1} + \binom{n}{4}+\binom{n}{7}+\cdots \]

and similar sum.  Considering problem 2.14, if you understand the solution, then you see it is the same idea as the proof of the identity you mentioned.  We use the fact that $\omega^2 + \omega + 1=0$ and $\omega^3=1$.  For the binomial expansion, instead of expanding $(1+x)^n$, what if you expand $x(1+x)^n$?