Perrin number

In mathematics, the Perrin numbers are defined by the recurrence relation

with initial values

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.

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).

The generating function of the Perrin sequence is

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

This formula can be used to quickly calculate values of the Perrin sequence for large n. The ratio of successive terms in the Perrin sequence approaches p, a.k.a. the plastic number, which has a value of approximately 1.324718. This constant bears the same relationship to the Perrin sequence as the golden ratio does to the Lucas sequence. Similar connections exist also between p and the Padovan sequence, between the golden ratio and Fibonacci numbers, and between the silver ratio and Pell numbers.

From the Binet formula, we can obtain a formula for G(kn) in terms of G(n−1), G(n) and G(n+1); we know

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