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

Author

Miklos Kristof, Jul 10 2002

Keywords

• nonn

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)).

Examples

			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.
		

Links

• Dan Kalman and Robert Mena, The Fibonacci Numbers: Exposed, Math. Mag. 76 (3) (2003) 167-181.
• Index to sequences related to cyclotomic polynomials.

Crossrefs

Cf. A000129, A002203.

Programs

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 *)

Extensions

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