Pell Number
The Pell numbers are the numbers obtained by the U_ns in the Lucas sequence with P=2 and Q=-1. They correspond to the Pell polynomial P_n(x) and Fibonacci polynomial F_n(x) values
The nth Pell number is therefore given in the Wolfram Language as Fibonacci [n, 2].
For n=0, 1, ..., the Pell numbers P_n are 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, ... (OEIS A000129). Note however that the alternate indexing convention P_0=1, P_1=2, ... are also used by some authors (e.g., Munarini 2019, Došlić and Podrug 2023), as is the alternate notational convention p_n (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
| P_n=2P_(n-1)+P_(n-2) |
(3)
|
with initial conditions P_0=0 and P_1=1 for the Pell numbers and Q_0=Q_1=2 for the Pell-Lucas numbers.
The generating function for the Pell numbers is
and so, by plugging in x=1/10,
The nth Pell number is explicitly given by the Binet-type formula
It is also given by the binomial sum
The Pell numbers satisfy the identities
See also
Brahmagupta Polynomial, Pell-Lucas Number, Pell Polynomial, Pell PrimeExplore with Wolfram|Alpha
More things to try:
References
Došlić, T. and Podrug, L. "Metallic Cubes." 26 Jul 2023. https://arxiv.org/abs/2307.14054.McDaniel, W. L. "Triangular Numbers in the Pell Sequence." Fib. Quart. 34, 105-107, 1996.Munarini, E. "Pell Graphs." Disc. Math. 342, 2415-2428, 2019.Ram, R. "Pell Numbers Formulae." http://users.tellurian.net/hsejar/maths/pell/.Sloane, N. J. A. Sequence A000129/M1413 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Pell NumberCite this as:
Weisstein, Eric W. "Pell Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PellNumber.html