Questions on the Lucas numbers, a special sequence of integers that satisfy the recurrence $L_n=L_{n-1}+L_{n-2}$ with the initial conditions $L_0=2$ and $L_1=1$.

The $n$th Lucas number $L_n$ is defined recursively, by

$$L_n = L_{n - 1} + L_{n - 2}$$

for $n > 1$, and $L_0 = 2,\; L_1 = 1$. There is a closed form expression, namely

$$ L_{n}=\varphi ^{n}+(1-\varphi )^{n}=\varphi ^{n}+(-\varphi )^{-n}=\left({1+{\sqrt {5}} \over 2}\right)^{n}+\left({1-{\sqrt {5}} \over 2}\right)^{n},$$

where the golden ratio $\varphi$ is equal to $\dfrac{1 + \sqrt{5}}{2}$.

The sequence of Lucas numbers is: $2,1,3,4,7,11,18,29,47,76,123,\dots$ (sequence A000032 in the OEIS).

Reference: Lucas number.