Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Lucas sequence
Summary
In mathematics, the Lucas sequences and are certain constant-recursive integer sequences that satisfy the recurrence relation
where and are fixed integers. Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences and
More generally, Lucas sequences and represent sequences of polynomials in and with integer coefficients.
Famous examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, Pell numbers, Lucas numbers, Jacobsthal numbers, and a superset of Fermat numbers (see below). Lucas sequences are named after the French mathematician Édouard Lucas.
Given two integer parameters and , the Lucas sequences of the first kind and of the second kind are defined by the recurrence relations:
and
It is not hard to show that for ,
The above relations can be stated in matrix form as follows:
Initial terms of Lucas sequences and are given in the table:
The characteristic equation of the recurrence relation for Lucas sequences and is:
It has the discriminant and the roots:
Thus:
Note that the sequence and the sequence also satisfy the recurrence relation. However these might not be integer sequences.
When , a and b are distinct and one quickly verifies that
It follows that the terms of Lucas sequences can be expressed in terms of a and b as follows
The case occurs exactly when for some integer S so that . In this case one easily finds that
The ordinary generating functions are
When , the Lucas sequences and satisfy certain Pell equations:
For any number c, the sequences and with
have the same discriminant as and :
For any number c, we also have
The terms of Lucas sequences satisfy relations that are generalizations of those between Fibonacci numbers and Lucas numbers . For example:
Among the consequences is that is a multiple of , i.e., the sequence
is a divisibility sequence. This implies, in particular, that can be prime only when n is prime.
Another consequence is an analog of exponentiation by squaring that allows fast computation of for large values of n.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.