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A072280
Product representation of the Pell numbers A000129 and A002203.
5
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
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
Sequence in context: A246751 A295850 A078104 * A217106 A329995 A086054
KEYWORD
nonn
AUTHOR
Miklos Kristof, Jul 10 2002
EXTENSIONS
Edited and extended by R. J. Mathar, Nov 27 2009
STATUS
approved