A general binomial

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

Given the binomial expansion above, evaluate the value of

2[(n0)+(n3)+(n6)++(nn)]+(1+ω)[(n1)+(n4)+(n7)++(nn2)]+(1+ω2)[(n2)+(n5)+(n8)++(nn1)]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

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