Search: id:A072280
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%I A072280
%S A072280 2,1,7,6,41,5,239,34,199,29,8119,33,47321,169,961,1154,1607521,197,9369319,
%T A072280 1121,32641,5741,318281039,1153,45245801,33461,7761799,38081,63018038201,
%U A072280 1345,367296043199,1331714,37667521,1136689,1273319041,39201,72722761475561
%N A072280 Product representation of the Pell numbers A000129 and A002203.
%C A072280 Define the silver mean constants h=1+sqrt(2) = A014176, h^2=1+2h = A156035, 
               and 1/h=h-2.
%C A072280 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 A072280 The Binet representations of the Pell numbers yields:
%C A072280 For even n, A000129(n) = Product_{d|n} a(d).
%C A072280 For odd n, A000129(n)=Product_{ d|n} a(2d).
%C A072280 For odd prime p, a(p)=A002203(p)/2, a(2p)=A000129(p).
%C A072280 a(2^(k+1))=A002203(2^k).
%C A072280 For odd n, A002203(n)=Product_{ d|n} a(d).
%C A072280 For k>0 and odd n, A002203(n*2^k)=Product_{ d | n} a(d*2^(k+1)).
%H A072280 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 A072280 <a href="Sindx_Cy.html#CYCLOTOMICPOLYNOMIALS">Index to sequences related 
               to cyclotomic polynomials</a>.
%e A072280 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 A072280 For odd n=9, A000129(9) = a(2)*a(6)*a(18)= 1*5*197 = 985.
%e A072280 For even n=8, A002203(12) = a(8)*a(24)=34*1153 = 39202.
%e A072280 For odd n=21, A002203(21) = a(1)*a(3)*a(7)*a(21) = 2*7*239*32641 = 109216786.
%p A072280 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 A072280 seq(A072280(n),n=1..80) ; # R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Nov 27 2009
%Y A072280 Cf. A000129, A002203.
%Y A072280 Adjacent sequences: A072277 A072278 A072279 this_sequence A072281 A072282 
               A072283
%K A072280 nonn,new
%O A072280 1,1
%A A072280 M. Kristof (kristmikl(AT)freemail.hu), Jul 10 2002
%E A072280 Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 
               27 2009

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