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).
For a Pell number 
to be prime, it is necessary that 
be prime. The indices of (probable) prime Pell numbers are
2, 3, 5, 11, 13, 29, 41, 53, 59, 89, 97, 101, 167, 181, 191, 523, 929, 1217, 1301,
1361, 2087, 2273, 2393, 8093, 13339, 14033, 23747, 28183, 34429, 36749, 90197, ...
(OEIS A096650), with no others less than 
(E. W. Weisstein, Mar. 21,
2009). The largest proven prime has index 13339 and 5106 digits (http://primes.utm.edu/primes/page.php?id=24572),
whereas the largest known probable prime has index 90197 and 34525 digits (T. D. Noe,
Sep. 2004).
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 
th Pell number is explicitly given by
the Binet-type formula
 |
(4)
|
The 
th Pell number is given by the binomial
sum
 |
(5)
|
The Pell numbers satisfy the identities
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
