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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

P_n=P_n(1)
(1)
=F_n(2).
(2)

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).

For a Pell number P_n to be prime, it is necessary that n 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 188856 (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

 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 nth Pell number is explicitly given by the Binet-type formula

 P_n=((1+sqrt(2))^n-(1-sqrt(2))^n)/(2sqrt(2)).
(4)

The nth Pell number is given by the binomial sum

 P_n=sum_(k=0)^(|_(n-1)/2_|)(n; 2k+1)2^k.
(5)

The Pell numbers satisfy the identities

P_(m+n)=P_mP_(n+1)+P_(m-1)P_n
(6)
P_(m+n)=2P_mQ_n-(-1)^nP_(m-n)
(7)
P_(m·2^t)=P_mproduct_(j=0)^(t-1)Q_(m·2^j).
(8)

See also

Brahmagupta Polynomial, Integer Sequence Primes, Pell-Lucas Number, Pell Polynomial

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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/.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 53-57, 1996.Sloane, N. J. A. Sequences A000129/M1413 and A096650 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Pell Number

Cite this as:

Weisstein, Eric W. "Pell Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PellNumber.html

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