The integer sequence defined by the recurrence
 |
(1)
|
with the initial conditions 
, 
, 
. The terms of the Perrin sequence are known as Perrin
numbers. This recurrence relation is the
same as that for the Padovan sequence but with
different initial conditions. The first few terms for 
, 1, ..., are 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, ... (OEIS
A001608).
The above cartoon (Amend 2005) shows an unconventional sports application of the Perrin sequence (right panel). (The left two panels instead apply the Fibonacci numbers).
is the solution of a third-order linear
homogeneous recurrence equation having characteristic equation
 |
(2)
|
Denoting the roots of this equation by 
, 
, and 
, with 
the unique real root, the solution is then
 |
(3)
|
Here,
 |
(4)
|
is the plastic constant 
, which is also given by the limit
 |
(5)
|
The asymptotic behavior of 
is
 |
(6)
|
Perrin sequence numbers that are prime are known as Perrin primes.
Perrin (1899) investigated the sequence and noticed that if 
is prime, then 
(i.e., 
divides 
). The first statement of this fact is attributed to É. Lucas
in 1876 by Stewart (1996). Perrin also searched for but did not find any composite
number 
in the sequence such that 
. Such numbers are now known as Perrin
pseudoprimes. Malo (1900), Escot (1901), and Jarden (1966) subsequently investigated
the series and also found no Perrin pseudoprimes.
Adams and Shanks (1982) subsequently found that 
is such a number.
