Integer powers for an annihilating polynomial with roots one and minus one ($(A-I)(A+I{)}^2=0$)

If a square matrix satisfies $(A-I)(A+I{)}^2=0$, then $A^{-1}=A^2+A-I$, and every integer matrix power lies in the span of $I,A,A^2$. Writing $s=(-1{)}^n$, the universal formula is $A^n={\alpha }_n{A}^2+{\beta }_{n}A+{\gamma }_{n}I$, where ${\alpha }_n=(1-s)/4+ns/2$, ${\beta }_n=(1-s)/2$ and ${\gamma }_n=(1+3s)/4-ns/2$. The polynomial $p_n(t)$ on the right satisfies $p_n(1)=1$, $p_n(-1)=(-1{)}^n$ and $p_n^{\prime }(-1)=n(-1{)}^{n-1}$. These value and derivative conditions implement interpolation at a repeated root. For negative integers the inverse formula first puts $A^n$ in the same polynomial algebra; the nilpotent block identity $(I-N{)}^n=I-nN$ for $N^2=0$ proves the derivative rule for every integer.