Fibonacci recurrence ($F_{n+1}=F_n+F_{n-1}$)

The linear recurrence relation defining the Fibonacci numbers, with $F_0=0$ and $F_1=1$. Its state $(F_{n+1},F_n{)}^T$ advances by the matrix $\left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right)$, so matrix powers describe repeated updates.