The Lucas numbers are related to the Fibonacci numbers by the identities
L n = F n − 1 + F n + 1 = F n + 2 F n − 1 = F n + 2 − F n − 2 {\displaystyle \,L_{n}=F_{n-1}+F_{n+1}=F_{n}+2F_{n-1}=F_{n+2}-F_{n-2}}
L m + n = L m + 1 F n + L m F n − 1 {\displaystyle \,L_{m+n}=L_{m+1}F_{n}+L_{m}F_{n-1}}
L n 2 = 5 F n 2 + 4 ( − 1 ) n {\displaystyle \,L_{n}^{2}=5F_{n}^{2}+4(-1)^{n}} , and thus as n {\displaystyle n\,} approaches +∞, the ratio L n F n {\displaystyle {\frac {L_{n}}{F_{n}}}} approaches 5 . {\displaystyle {\sqrt {5}}.}
F 2 n = L n F n {\displaystyle \,F_{2n}=L_{n}F_{n}}
F n + k + ( − 1 ) k F n − k = L k F n {\displaystyle \,F_{n+k}+(-1)^{k}F_{n-k}=L_{k}F_{n}}
F n = L n − 1 + L n + 1 5 = L n − 3 + L n + 3 10 {\displaystyle \,F_{n}={L_{n-1}+L_{n+1} \over 5}={L_{n-3}+L_{n+3} \over 10}}