A072270 A partial product representation of A006131 and A072265.

1, 1, 13, 9, 101, 5, 701, 49, 361, 29, 31021, 33, 204101, 181, 1021, 1889, 8799541, 233, 57746701, 1361, 41581, 7589, 2486401661, 1633, 161532401, 49661, 22810681, 58241, 702418373381, 2245, 4608956945501, 3437249, 74991181, 2135149, 2802699901, 75921, 1302034904649701, 14007941, 3219888061, 3019201

Offset

1,3

Comments

Define f(n) = A006131(n-1) and L(n) = 4*f(n-1)+f(n+1), which implies L(n) = A072265(n), n>1.

For even n, f(n) = product_{d|n} a(d) and for odd n, f(n) = product_{d|n} a(2d).

Taking logarithms defines the sequence a(.) via a Mobius transformation (see A072183).

Writing f(n) and L(n) in terms of Binet formulas leads to a representation as cyclotomic polynomials.

Links

Table of n, a(n) for n=1..40.

Formula

Let h=(1+sqrt(17))/2, Phi(n, x) = n-th cyclotomic polynomial, so x^n-1= product_{d|n} Phi(d, x), and let g(d) be the order of Phi(d, x). Then a(n)=(h-1)^g(n)*Phi(n, h^2/4), n>2.

a(p) = L(p) for odd prime p.

a(2p) = f(p) for odd prime p.

a(2^k+1) = L(2^k).

a(3*2^k = L(2^k)-4^k.

L(n) = product_{d|n} a(d) for odd n.

L(n*2^k) = product_{d|n} a(d*2^(k+1)) for k>0 and odd n.

Example

f(12)=a(1)*a(2)*a(3)*a(4)*a(6)*a(12) = 1*1*13*9*5*33 = 19305 for even n=12.
f(9)=a(2)*a(6)*a(18)= 1*5*233 = 1165 for odd n=9.
L(6)=a(4)*a(12) = 9*33 = 297 = 4*f(5)+f(7) = 4*29+181 for even n=6.
L(15)=a(1)*a(3)*a(5)*a(15) = 1*13*101*1021 = 1340573 for odd n=15.

Maple

A072270 := proc(n) if n <=2 then 1; else h := (1+sqrt(17))/2 ; cy := numtheory[cyclotomic](n, x) ; g := degree(cy) ; (h-1)^g*subs(x=h^2/4, cy) ; expand(%) ; end if; end proc: # R. J. Mathar, Nov 17 2010

Crossrefs

Cf. A006131, A072183.

Sequence in context: A066552 A206609 A281085 * A214025 A240812 A291425

Adjacent sequences: A072267 A072268 A072269 * A072271 A072272 A072273

Keyword

nonn

Author

Miklos Kristof, Jul 09 2002

Extensions

Divided argument of Phi by 4; moved comments to formula section - R. J. Mathar, Nov 17 2010

Status

approved