A072280 - OEIS

%I #12 May 08 2023 09:33:17

%S 2,1,7,6,41,5,239,34,199,29,8119,33,47321,169,961,1154,1607521,197,

%T 9369319,1121,32641,5741,318281039,1153,45245801,33461,7761799,38081,

%U 63018038201,1345,367296043199,1331714,37667521,1136689,1273319041,39201,72722761475561

%N Product representation of the Pell numbers A000129 and A002203.

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

%C 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.

%C The Binet representations of the Pell numbers yields:

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

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

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

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

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

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

%H Dan Kalman and Robert Mena, <a href="http://www.jstor.org/stable/3219318">The Fibonacci Numbers: Exposed</a>, Math. Mag. 76 (3) (2003) 167-181.

%H <a href="/index/Cy#CYCLOTOMICPOLYNOMIALS">Index to sequences related to cyclotomic polynomials</a>.

%e 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.

%e For odd n=9, A000129(9) = a(2)*a(6)*a(18)= 1*5*197 = 985.

%e For even n=8, A002203(12) = a(8)*a(24)=34*1153 = 39202.

%e For odd n=21, A002203(21) = a(1)*a(3)*a(7)*a(21) = 2*7*239*32641 = 109216786.

%p 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:

%p seq(A072280(n),n=1..80) ; # _R. J. Mathar_, Nov 27 2009

%t a[n_] := If[n <= 2, 3-n, g = EulerPhi[n]; h = 1 + Sqrt[2]; (h - 2)^g*Cyclotomic[n, h^2] // Expand];

%t Table[a[n], {n, 1, 80}] (* _Jean-François Alcover_, May 08 2023, after _R. J. Mathar_ *)

%Y Cf. A000129, A002203.

%K nonn

%O 1,1

%A _Miklos Kristof_, Jul 10 2002

%E Edited and extended by _R. J. Mathar_, Nov 27 2009