Combinatory Analisis and Newton's Binomial

Question

Why is $C_{n}^{0} \ + \ C_{n}^{3} \ +\ C_{n}^{6}\ +\ ...=\ \frac13\cdot[2^n + 2\cdot\cos({\frac{n\pi}3})]$?

Answer 1

If you know Newton's binomial expansion formula: $$(1 + x)^n = \sum_{i = 0}^nC_n^ix^i,$$ then you just apply it to $x = 1, \omega, \omega^2$, where $\omega = \frac{-1 + \sqrt3 i}2$ is a third root of unity.

Adding up the three identities thus obtained (and then dividing by $3$), you will get what you stated in the question.