The Lucas sequence is an infinite series of integers that recursively approximate the golden ratio and is linearly related to the Fibonacci number series.
In other words, the Lucas sequence is a series of numbers that, by adding or subtracting, approximate an irrational number called the golden ratio and is very similar to the Fibonacci series.
Lucas’s succession
Since it is an infinite series, in the following table we will only show the first sixteen numbers. To know any other number in the series, simply apply the following function. The Lucas series is a progression in which each number is obtained from the addition or subtraction of the previous or subsequent number respectively.
| Index (i) | Lucas series (L I ) | Index (i) | Lucas series (L I ) |
| one | two | 9 | 47 |
| two | one | 10 | 76 |
| 3 | 3 | eleven | 123 |
| 4 | 4 | 12 | 199 |
| 5 | 7 | 13 | 322 |
| 6 | eleven | 14 | 521 |
| 7 | 18 | fifteen | 843 |
| 8 | 29 | 16 | 1364 |
Function for the succession of Lucas
Function for the succession of Lucas
Where L represent the series numbers and the subscript i the position within the series, then, if we want to represent the fifth number in the series, we will represent it as L5.
In other words, depending on whether we want to get the next or previous number in the series, we add or subtract, for example:
2 + 1 = 3 18 – 11 = 7
1 + 3 = 4 11 – 7 = 4
Representation of the succession of Lucas
Representation of the first numbers of the Fibonacci series.
History
The creator of this number series is F. Édouard A. Lucas, a French mathematician who, apart from working with the Fibonacci series, also created a very famous game called the Towers of Hanoi.
Application
The Lucas series is not very well known since all the importance has been taken by the Fibonacci series. Many people only relate the golden ratio to the Fibonacci series when in reality both series approach it. We can also find Lucas patterns in some objects and elements of nature.
