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Given the roots of $x^3=x^2+1$, we have sequence A001609,

$M(n) = x_1^n+x_2^n+x_3^n = \,_3F_2\left(\frac{-n}{3}, \frac{1-n}{3}, \frac{2-n}{3};\; \frac{1-n}{2}, \frac{2-n}{2};\; -\frac{3^3}{2^2}\right) = 1, 1, 4, 5, 6, 10, 15, 21,\dots$

for $n = {1,2,3,\dots}$

Question: Given $y^3=y+1$, is there any similar generalized hypergeometric formula for the Perrin numbers?

$P(n) = y_1^n+y_2^n+y_3^n = 0,2,3,2,5,5,7,10,\dots$

The closest I found is the binomial sum,

$ \begin{aligned}P(n) &= n\sum_{k=1}^{n/2} \frac{\binom k{n-2k}}{k} = 0,2,3,2,5,5,7,10,\dots\end{aligned}$

where both start with $n = 1,2,3,\dots$ Anyone knows how to translate that into the generalized hypergeometric function?

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