Lucas number

The sequence of Lucas numbers is strongly related to the sequence of Fibonacci numbers. Lucas number and Fibonacci numbers have the identical formula $a_{n}=a_{n-1}+a_{n-2}\$, and both sequences are part of the Lucas sequence with the parameter P=1 and Q=(-1).

$L_{n}:=L(n):={\begin{cases}2&{\mbox{if }}n=0;\\1&{\mbox{if }}n=1;\\L(n-1)+L(n-2)&{\mbox{if }}n>1.\\\end{cases}}$

The first few Lucas numbers are: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ...

Properties

• If $p\$ is a prime number, than $p\$ divides $L_{p}-1\$. The converse is false.

• Relationship to the Fibonacci number is given by $L_{n}=F_{n-1}+F_{n+1}\$.