Definition:Lucas Number/Definition 1

Definition

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}$

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 start the sequence at $1$:

$L_n = \begin{cases}

1 & : n = 1 \\ 3 & : n = 2 \\ L_{n - 1} + L_{n - 2} & : \text{otherwise} \end{cases}$

Also see

• Equivalence of Definitions of Lucas Numbers

Source of Name

This entry was named for François Édouard Anatole Lucas.

Sources

• 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction: Example $1 \text{-} 1$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $11$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Lucas numbers
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $11$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Lucas numbers