Perrin number
In mathematics, the Perrin numbers are defined by the recurrence relation
and
The sequence of Perrin numbers starts with
The number of different maximal independent sets in an n-vertex cycle graph is counted by the nth Perrin number for n > 1.
History
This sequence was mentioned implicitly by Édouard Lucas (1876). In 1899, the same sequence was mentioned explicitly by François Olivier Raoul Perrin. The most extensive treatment of this sequence was given by Adams and Shanks (1982).
Properties
Generating function
The generating function of the Perrin sequence is
Matrix formula
Binet-like formula
The Perrin sequence numbers can be written in terms of powers of the roots of the equation
This equation has 3 roots; one real root p (known as the plastic number) and two complex conjugate roots q and r. Given these three roots, the Perrin sequence analogue of the Lucas sequence Binet formula is
Since the magnitudes of the complex roots q and r are both less than 1, the powers of these roots approach 0 for large n. For large n the formula reduces to
Podcasts:
