Equivalence of Definitions of Lucas Numbers

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.

$\blacksquare$