No use of any identities

Algebra Level 4

Let α,β\alpha,\beta be the roots of the equation x2x1=0x^2-x-1=0.

Let a2n=α2n+β2na_{2n}={\alpha}^{2n}+{\beta}^{2n} where n1n\geq1 and nn is a natural number.

What is the value of a20+a22a_{20}+a_{22} ?

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