Question

What is the difference between a Fibonacci sequence and a Lucas sequence?

Answer 1

The Fibonacci numbers are a particular example of a Lucas sequence.

Answer 2

Can you be a bit more specific, something like typing down the numbers will be better for me to understand. ..

Answer 3

A Lucas sequence is defined as follows:\[x_n=Px_{n-1}+Qx_{x-2}\text{, where }P,Q\in\mathbb{Z}.\]
The Fibonacci sequence has it that \(P=Q=1\) and is as follows:\[1,1,2,3,5,8,13,\dots\]However, What if I were to change \(P=Q=2\)? Then I would get:\[1,1,4,10,28,76,\dots\]which is not the Fibonacci sequence but it is a Lucas sequence.