Lucas number ($L_0=2,L_1=1,L_{n+2}=L_{n+1}+L_n$)

The Lucas numbers obey the same second-order recurrence as the Fibonacci numbers but start with $2,1$, so their sequence is $2,1,3,4,7,11,\dots$. For $n\ge 1$, $L_n=F_{n-1}+F_{n+1}$. Substitution in the Fibonacci addition formula gives $F_{2n}=F_n{L}_n$. Consecutive Fibonacci numbers are coprime by the Euclidean algorithm; since $L_n=F_n+2F_{n-1}$, Bézout's identity gives $\gcd (F_n,L_n)=\gcd (F_n,2)\le 2$.

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 ${\psi }^n$.