No use of any identities

Algebra Level 5

Let $\alpha,\beta$ be the roots of the equation $x^2-x-1=0$ .

Let $a_{2n}={\alpha}^{2n}+{\beta}^{2n}$ where $n\geq1$ and $n$ is a natural number.

What is the value of $a_{20}+a_{22}$ ?