A072280 Product representation of the Pell numbers A000129 and A002203.

2, 1, 7, 6, 41, 5, 239, 34, 199, 29, 8119, 33, 47321, 169, 961, 1154, 1607521, 197, 9369319, 1121, 32641, 5741, 318281039, 1153, 45245801, 33461, 7761799, 38081, 63018038201, 1345, 367296043199, 1331714, 37667521, 1136689, 1273319041, 39201, 72722761475561

Offset

1,1

Comments

Define the silver mean constants h=1+sqrt(2) = A014176, h^2=1+2h = A156035, and 1/h=h-2.

Let Phi(n,x) be the n-th cyclotomic polynomial A013595, so that x^n-1 = Product_{d | n} Phi(d, x). Let g(n) be the order of Phi(n, x), A000010. Then a(n)=(h-2)^g(n)*Phi(n, h^2) if n <> 2.

The Binet representations of the Pell numbers yields:

For even n, A000129(n) = Product_{d|n} a(d).

For odd n, A000129(n)=Product_{ d|n} a(2d).

For odd prime p, a(p)=A002203(p)/2, a(2p)=A000129(p).

a(2^(k+1))=A002203(2^k).

For odd n, A002203(n)=Product_{ d|n} a(d).

For k>0 and odd n, A002203(n*2^k)=Product_{ d | n} a(d*2^(k+1)).

Links

Table of n, a(n) for n=1..37.

Dan Kalman and Robert Mena, The Fibonacci Numbers: Exposed, Math. Mag. 76 (3) (2003) 167-181.

Index to sequences related to cyclotomic polynomials.

Example

For even n=12, A000129(12) = a(1)*a(2)*a(3)*a(4)*a(6)*a(12) = 2*1*7*6*5*33 = 13860.
For odd n=9, A000129(9) = a(2)*a(6)*a(18)= 1*5*197 = 985.
For even n=8, A002203(12) = a(8)*a(24)=34*1153 = 39202.
For odd n=21, A002203(21) = a(1)*a(3)*a(7)*a(21) = 2*7*239*32641 = 109216786.

Maple

A072280 := proc(n) if n <= 2 then 3-n ; else g := numtheory[phi](n) ; h := 1+sqrt(2) ; (h-2)^g*numtheory[cyclotomic](n, h^2) ; simplify(expand(%)) ; end if; end proc:
seq(A072280(n), n=1..80) ; # R. J. Mathar, Nov 27 2009

Mathematica

a[n_] := If[n <= 2, 3-n, g = EulerPhi[n]; h = 1 + Sqrt[2]; (h - 2)^g*Cyclotomic[n, h^2] // Expand];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, May 08 2023, after R. J. Mathar *)

Crossrefs

Cf. A000129, A002203.

Sequence in context: A246751 A295850 A078104 * A217106 A329995 A086054

Adjacent sequences: A072277 A072278 A072279 * A072281 A072282 A072283

Keyword

nonn

Author

Miklos Kristof, Jul 10 2002

Extensions

Edited and extended by R. J. Mathar, Nov 27 2009

Status

approved