In
mathematics, the Perrin numbers are defined by the
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a paramete ...
: for ,
with initial values
:.
The sequence of Perrin numbers starts with
:
3,
0,
2, 3, 2,
5, 5,
7,
10,
12,
17,
22,
29,
39, ...
The number of different
maximal independent set
In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other words, there is no vertex outside the independent set that may join it because it is maxim ...
s in an -vertex
cycle graph
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with vertices is called ...
is counted by the th Perrin number for .
History
This sequence was mentioned implicitly by
Édouard Lucas
__NOTOC__
François Édouard Anatole Lucas (; 4 April 1842 – 3 October 1891) was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him.
Biography
Lu ...
(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
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary ser ...
of the Perrin sequence is
:
Matrix formula