 Lucas number

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 $\gcd (F_n,L_n)=\gcd (F_n,2)\le 2$.

Lucas numbers are a sequence of numbers that are similar to the Fibonacci numbers but start with different initial values. The Lucas sequence is defined as follows: 1. The first two terms of the sequence are \(L_0 = 2\) and \(L_1 = 1\).