Recursion of S_n=x_1^n + x_2^n + x_3^n

Recursion of ${S}_{n}={x}_{1}^{n}+{x}_{2}^{n}+{x}_{3}^{n}$
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Reinfarktq6
${x}_{1},{x}_{2},{x}_{3}$ are roots of $P\left(x\right)={x}^{3}-\alpha {x}^{2}+\beta x-\gamma$
Then, plugging in ${x}_{i}$ and multiplying by ${x}_{i}^{n}$ you get
${x}_{i}^{n+3}-\alpha {x}_{i}^{n+2}+\beta {x}_{i}^{n+1}-\gamma {x}_{i}^{n}=0\phantom{\rule{thinmathspace}{0ex}}.$
Adding the three relations we get the relation Thomas Andrews wrote:
${S}_{n+3}-\alpha {S}_{n+2}+\beta {S}_{n+1}-\gamma {S}_{n}=0\phantom{\rule{thinmathspace}{0ex}}.$