The Perrin sequence is the integer sequence defined by the recurrence relation
 |
(1)
|
for 
, with 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 index 
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.
