Lucas closed form ($L_n={\phi }^n+{\psi }^n$)

For $\phi =(1+\sqrt{5})/2$ and $\psi =(1-\sqrt{5})/2$, each power sequence satisfies the recurrence relation $x_{n+2}=x_{n+1}+x_n$, because both bases solve $t^2=t+1$. Their sum starts at $2,1$, so uniqueness of a recursively defined sequence proves that it is the Lucas number sequence. Alternatively, diagonalize the companion matrix $\left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right)$ and use $(1,2{)}^T=(\phi ,1{)}^T+(\psi ,1{)}^T$. The formula explains the exponential growth governed by the golden ratio and the alternating decaying correction ${\psi }^n$.