# Equivalence of Definitions of Lucas Numbers

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## Theorem

The following definitions of the concept of **Lucas Number** are equivalent:

### 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.

## Proof

### Definition 1 implies Definition 2

Let $\sequence {L_n}$ be the sequence defined as in definition 1.

It follows from Lucas Number as Sum of Fibonacci Numbers that $\sequence {L_n}$ is the sequence defined as in definition 2.

$\Box$

### Definition 2 implies Definition 1

Let $\sequence {L_n}$ be the sequence defined as in definition 2.

It follows from Lucas Number as Element of Recursive Sequence that $\sequence {L_n}$ is the sequence defined as in definition 1.

$\blacksquare$