The Pell numbers are the numbers obtained by the 
s in the Lucas sequence
with 
and 
. They correspond to the Pell
polynomial 
and Fibonacci polynomial 
values
 |  |  |
(1)
|
 |  |  |
(2)
|
The 
th Pell number is therefore given in the
Wolfram Language as Fibonacci[n,
2].
For 
, 1, ..., the Pell numbers 
are 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, ... (OEIS
A000129). Note however that the alternate indexing
convention 
,
, ... are also used by some authors
(e.g., Munarini 2019, Došlić and Podrug 2023), as is the alternate notational
convention 
(e.g., Munarini 2019).
The only triangular Pell number is 1 (McDaniel 1996). Pell numbers that are prime are known as Pell primes.
The Pell and Pell-Lucas numbers satisfy the recurrence relation
 |
(3)
|
with initial conditions 
and 
for the Pell numbers and 
for the Pell-Lucas
numbers.
The generating function for the Pell numbers is
 |
(4)
|
and so, by plugging in 
,
 |
(5)
|
The 
th Pell number is explicitly given by
the Binet-type formula
 |
(6)
|
It is also given by the binomial sum
 |
(7)
|
The Pell numbers satisfy the identities
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
