Find all integer $n$ such that $x^2+x+1$ divides $X^{2n}+(x+1)^{2n}+1$

Question

I tried particular cases, with $n=1,2,3,4$... The pattern I find is that when $n$ is multiple of 3 it doen't divide. I think is related with the binomial theorem.

Answer 1

$$x^{2n}+(x+1)^{2n}+1\equiv x^{2n}+(-x^2)^{2n}+1=x^{4n}+x^{2n}+1.$$ Now, use that $x^3\equiv1.$

I got that should be $n$ is a positive integer and $n$ is not divisible by $3$.