Fibonacci and Lucas
Abstract
The Fibonacci sequence is based on a linear recurrence equation ((p,q)=(1,1)) and has a special the property: symmetry around 0. This means that the absolute value for Fn and F-n are equal. It appears that only sequences that begin with (0,1) or (2/p,1) are symmetric. The Lucas sequence is an example of the last one (p=1).
Both sequences (Fibonacci and Lucas) have the property that the ratio of the last two values tends to a constant: the golden ratio, which is defined as the Greek letter phi.
A possible solution of this linear recurrence equation is the power of phi. It appears that another possible solution equals minus the power of minus phi. This value will be defined as the Greek letter psi. The Fibonacci and Lucas sequences are based on phi and psi.
- 1 Symmetry
- 2 The Golden Ratio
- 3 Phi and psi
