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 immediately previous terms. However, unlike the Fibonacci numbers, which start as \(1, 2, ...,\) Lucas numbers start as \(2, 1, ...\).

The first few Lucas numbers are as follows: \(2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ...\).

As a recurrence relation, Lucas numbers are defined as

\[L_0=2, L_1 = 1, L_2 = 3, \dots, L_n = L_{n-2} + L_{n-1}.\]

The ratio between two consecutive Lucas numbers converges to the golden ratio \(1.61803398875\ldots\).Which means \(\displaystyle \lim_{n \to \infty} \frac{\lfloor \phi^{n+1} \rfloor}{\lfloor \phi^n \rfloor}=\phi\)

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, \(L_n = \lfloor \phi^{n} \rfloor,\) where \(\lfloor \cdot \rfloor\) is the floor function and \(L_n\) is the \(n^\text{th}\) Lucas number, but this function does not work for 1. This also means that the Lucas numbers can be derived from the golden ratio itself.

Finding Lucas Numbers

What is the \(11^\text{th}\) Lucas number?

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We have \(L_1 = 1, L_2 = 3, \ldots, L_n = L_{n-2} + L_{n-1};\) however, instead of working back, we can find it by using the \(L_n\) relationship to \(\phi\).

We know that

\[L_n = \lfloor \phi^{n} \rfloor.\]

So,

\[\begin{align} L_{11} &= \big\lfloor 1.61803^{11} \big\rfloor\\ &= \lfloor 199.005 \rfloor \\ &= 199. \end{align} \]

So, our final answer is \(199.\ _\square\)