What is the numerical value of this tricky equation?

[anemone]

Here is this week's POTW:

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If $x^2+x+1=0$, compute the numerical value of

\(\displaystyle \left(x+\frac{1}{x}\right)^2+\left(x^2+\frac{1}{x^2}\right)^2+\left(x^3+\frac{1}{x^3}\right)^2+\cdots+\left(x^{29}+\frac{1}{x^{29}}\right)^2\).

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[anemone]

Congratulations to the following members for their correct solution::)

1. greg1313
2. kaliprasad
3. johng

Solution from jonng:

Spoiler

Since $x^2+x+1=0$, $x^3=1\text{ but }x\neq1$. So ${1\over x}=x^2$. Now consider a term $(x^k+(x^{-1})^k)^2=(x^k+x^{2k})^2$. Now if $k\equiv 0\pmod{3}$, this term is 4; otherwise the term is $1$ since $x^k\neq 1$ is a cube root of 1 and hence satisfies $x^{2k}+x^k+1=0$. So there are 9 terms of the 29 that have value 4 and the remaining 20 have value 1. Thus the sum of the 29 terms is 56.