= Lucas closed form
{c}
{title2=$L_n=\varphi^n+\psi^n$}
For $\varphi=(1+\sqrt5)/2$ and $\psi=(1-\sqrt5)/2$, 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> $\bigl(\begin{smallmatrix}1&1\\1&0\end{smallmatrix}\bigr)$ and use $(1,2)^T=(\varphi,1)^T+(\psi,1)^T$. The formula explains the exponential growth governed by the <golden ratio> and the alternating decaying correction $\psi^n$.
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