Lucas sequence

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.