Perrin number
Introduction
The Perrin sequence, named after French mathematician François Perrin, is a sequence of integers that is defined in a similar manner to the Fibonacci sequence, but with a different initial three numbers. The sequence is defined recursively by the formula P(n) = P(n-2) + P(n-3) for n > 2, with initial values P(0) = 3, P(1) = 0, and P(2) = 2.
Mathematical Properties
The Perrin sequence has a number of interesting mathematical properties. For instance, it is a divisibility sequence, meaning that for any two positive integers m and n, if m divides n, then P(m) divides P(n). This property is shared by many other integer sequences, including the Fibonacci sequence.
The Perrin sequence also exhibits a property known as the Perrin pseudoprime property. A Perrin pseudoprime is a composite number n for which P(n) is divisible by n. This property is analogous to the Carmichael numbers in the Fibonacci sequence, which are composite numbers n for which F(n) is divisible by n.
In addition, the Perrin sequence has a closed-form formula, which is a formula that allows the computation of the nth term of the sequence without having to compute the preceding terms. This formula involves the roots of the cubic equation x^3 - x - 1 = 0, which are approximately 1.324717957244746, -0.662358978622373 - 0.562279512062301i, and -0.662358978622373 + 0.562279512062301i.
Applications
The Perrin sequence has applications in a number of areas of mathematics, including number theory, combinatorics, and graph theory. In number theory, the Perrin sequence is used in the study of pseudoprimes and primality testing. In combinatorics, the Perrin sequence appears in the enumeration of certain types of combinatorial structures. In graph theory, the Perrin sequence is related to the number of perfect matchings in certain types of graphs.