A general binomial

${(1+x)}^n = \displaystyle \sum_{r=0}^n \dbinom nr x^r$

Given the binomial expansion above, evaluate the value of

$2 \left[ \dbinom n0 + \dbinom n3 + \dbinom n6 + \cdots + \dbinom nn \right] \\ + (1 + \omega) \left[ \dbinom n1 + \dbinom n4 + \dbinom n7 + \cdots + \dbinom{n}{n-2} \right] \\ + (1 + \omega^2) \left[ \dbinom n2 + \dbinom n5 + \dbinom n8 + \cdots + \dbinom{n}{n-1} \right]$

where

• $\omega$ denotes the non-real cube root of unity.
• $n$ in an odd integral multiple of 3.
• $\dbinom MN = \dfrac {M!}{N! (M-N)!}$ denotes the binomial coefficient.