A127259 Sequence arising from the factorization of F(n)=A002605 and L(n)=A080040 F(0)=0, F(1)=1, F(n)=2*F(n-1)+2*F(n-2), L(0)=2, L(1)=2, L(n)=2*L(n-1)+2*L(n-2).

2, 1, 10, 8, 76, 6, 568, 56, 424, 44, 31648, 52, 236224, 328, 2320, 3104, 13160704, 408, 98232832, 2896, 129088, 18272, 5472827392, 3088, 537496576, 136384, 71911936, 161344, 2275853910016, 3856, 16987204845568

Offset

1,1

Links

Table of n, a(n) for n=1..31.

Formula

(sqrt(3)-1)^degree(cyclotomic(n,x),x)*cyclotomic(n,2+sqrt(3)) L(n)=2*F(n-1)+F(n+1) F(2n)=Product(d|2n) a(d), F(2n+1)=Product(d|2n+1) a(2d). L(2n+1)=Product(d|2n+1, a(d)), for k>0: L(2^k*(2n+1))=Product(d|2n+1, a(2^(k+1)*d)). for odd prime p, a(p)=L(p)/2, a(2p)=f(p) a(1)=2, a(2)=1; a(2^(k+1))=L(2^k);

Example

F(12)=a(1)*a(2)*a(3)*a(4)*a(6)*a(12)=2*1*10*8*6*52=49920
F(9)=a(2)*a(6)*a(18)= 1*6*408=2448
L(12)=a(8)*a(24)=56*3088=172928
L(21)=a(1)*a(3)*a(7)*a(21)=2*10*568*129088=1466439680

Maple

with(numtheory): a[1]:=2:a[2]:=1:for n from 3 to 60 do a[n]:=round(evalf((sqrt(3)-1)^degree(cyclotomic(n, x), x)*cyclotomic(n, 2+sqrt(3)), 30)) od: seq(a[n], n=1..60);

Crossrefs

Cf. A002605, A080040.

Sequence in context: A113088 A060694 A217108 * A152260 A286781 A193727

Adjacent sequences: A127256 A127257 A127258 * A127260 A127261 A127262

Keyword

nonn

Author

Miklos Kristof, Mar 26 2007

Status

approved