# Definition:Lucas Number

## Definition

### Definition 1

The Lucas numbers are a sequence which is formally defined recursively as:

$L_n = \begin{cases} 2 & : n = 0 \\ 1 & : n = 1 \\ L_{n - 1} + L_{n - 2} & : \text{otherwise} \end{cases}$

### Definition 2

The Lucas numbers are a sequence defined as:

$L_n = F_{n - 1} + F_{n + 1}$

where $F_k$ is the $k$th Fibonacci number.

### Sequence

The Lucas sequence begins:

$2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, \ldots$

## Also defined as

Some sources begin this sequence:

$1, 3, 4, 7, \ldots$

## Also see

• Results about Lucas numbers can be found here.

## Source of Name

This entry was named for François Édouard Anatole Lucas.