Lucas Numbers
The Lucas numbers are an integer sequence similar to the Fibonacci sequence, where each number is the sum of the two preceding ones. The sequence is named after the mathematician François Édouard Anatole Lucas.
Definition
The Lucas numbers are defined by the recurrence relation1:
\[
L_n = L_{n-1} + L_{n-2}
\]
with initial conditions:
\[
L_0 = 2,\quad L_1 = 1
\]
Closed-form Expression
The Lucas numbers have a closed-form expression using the golden ratio \(\phi\):
\[
L_n = \phi^n + (1 - \phi)^n
\]
where \(\phi = \frac{1 + \sqrt{5}}{2}\) is the golden ratio.
Relationship with Fibonacci Numbers
The Lucas numbers are closely related to Fibonacci numbers \(F_n\):
\[
L_n = F_{n-1} + F_{n+1}
\]
First Few Lucas Numbers
| n | Lₙ |
|---|---|
| 0 | 2 |
| 1 | 1 |
| 2 | 3 |
| 3 | 4 |
| 4 | 7 |
| 5 | 11 |
| 6 | 18 |
| 7 | 29 |
| 8 | 47 |
| 9 | 76 |