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: $2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ...$ whose construction is as follows:

Fibonacci adding

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 $1.61803398875\ldots$.

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, $L_n = \lfloor \phi^{n} \rceil,$ where $\lfloor \cdot \rceil$ is the nearest integer function for $n\geq 2 .$

We are given that $\left( \dfrac{1+ \sqrt5}2 \right)^{11}$ and $\left( \dfrac{1+ \sqrt5}2 \right)^{12}$ are equal to $199.005$ and $321.997,$ $($to $3$ decimal places$)$ respectively. What is the value of the $13^\text{th}$ Lucas number?

---

The result above tells us that

• $L_{11} = \left \lfloor \left( \dfrac{1+ \sqrt5}2 \right)^{11} \right \rceil = \lfloor 199.005 \rceil = 199 ,$ and

• $L_{12} = \left \lfloor \left( \dfrac{1+ \sqrt5}2 \right)^{12} \right \rceil = \lfloor 321.997 \rceil = 322 .$

Using the recursive formula of $L_n ,$ we get \[ L_{13} = L_{12} + L_{11} = 322 + 199 = {521} . \, _\square \]