1,1

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)).

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

Index to sequences related to cyclotomic polynomials.

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.

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 (mathar(AT)strw.leidenuniv.nl), Nov 27 2009

Cf. A000129, A002203.

nonn,new

M. Kristof (kristmikl(AT)freemail.hu), Jul 10 2002

Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 27 2009