Let α and β be two real numbers such that α+β=1 and α β=-1. Let p_n =(α)^n +(β)^n, p_n -1 =11 and p_n +1 =29 for some integer n ≥ 1. Then, the value…

Let $\displaystyle \alpha$ and $\displaystyle \beta$ be two real numbers such that $\displaystyle \alpha+\beta=1$ and $\displaystyle \alpha \beta=-1$. Let $\displaystyle \mathrm{p}_{\mathrm{n}}=(\alpha)^{\mathrm{n}}+(\beta)^{\mathrm{n}}$, $\displaystyle \mathrm{p}_{\mathrm{n}-1}=11$ and $\displaystyle \mathrm{p}_{\mathrm{n}+1}=29$ for some integer $\displaystyle \mathrm{n} \geqslant 1$. Then, the value of $\displaystyle \mathrm{p}_{\mathrm{n}}^2$ is \_\_\_\_.