Lucas Numbers

The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–91), who studied both that sequence and the closely related Fibonacci numbers.

Similar to the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate previous terms. However, unlike the Fibonacci numbers, which starts as \(1, 2 . . .\) Lucas number start as \(2, 1 . . .\)

The first few Lucas numbers:

\(2, 1, 3, 4, 7, 11, 18, 29, 47, 76 . . .\)

As a recurrence relation, Lucas numbers are defined as:

\(L_0 = 1, L_2 = 1 \ldots L_n = L_n-2 + L_n-1\)

The ratio between two consecutive Lucas numbers converges to the golden ratio, 1.61803398875. . .

A Lucas Number Spiral

The Golden Ratio is found in nature everywhere you look. It is obtained by dividing a line into two parts such that the longer part divided by the smaller part is also equal to the whole length divided by the longer part:

The golden ratio

It is approximated as \(1.61803\), and denoted as \(\phi\)

Interestingly enough, \(L_n\) = [\(\phi^{n-1}]\) where \([.]\) is the number rounded to the nearest whole. This means that the Lucas numbers can be derived from the Golden Ration itself.

Finding Lucas Numbers

What is the 11th Lucas number?

\(L_0 = 1, L_2 = 1 \ldots L_n = L_n-2 + L_n-1\), however, instead of working back, we can find it by using the \(L_n\) relationship to \(\phi\).

We know that

\[L_n = [\phi^{n-1}]\]

So,

\[L_{11} = [1.61803^{11-1}]\] \[L_{11} = [1.61803^{10}]\] \[L_{11} = [122.988] \] \[L_{11} = 123 \]

And our final answer is 123. \( _\square \)