= Lucas number
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{title2=$L_0=2,\ L_1=1,\ L_{n+2}=L_{n+1}+L_n$}
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The <Lucas numbers> obey the same second-order recurrence as the <Fibonacci numbers> but start with $2,1$, so their sequence is $2,1,3,4,7,11,\ldots$. For $n\ge1$, $L_n=F_{n-1}+F_{n+1}$. Substitution in the <Fibonacci addition formula> gives $F_{2n}=F_nL_n$. Consecutive <Fibonacci numbers> are coprime by the <Euclidean algorithm>; since $L_n=F_n+2F_{n-1}$, <Bézout's identity> gives $\gcd(F_n,L_n)=\gcd(F_n,2)\le2$.
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