A127609 Sequence arising from the factorization of F(n)= A091914(n-1) and L(n)= A127262. F(0)=0, F(1)=1, F(n)=2*F(n-1)+12*F(n-2), L(0)=2, L(1)=2, L(n)=2*L(n-1)+12*L(n-2).
2, 1, 40, 28, 976, 16, 21568, 496, 11584, 304, 9868288, 352, 209588224, 6208, 113920, 204544, 94347526144, 8128, 2001299832832, 153856, 49205248, 2747392, 900422667599872, 183808, 19568631218176, 58200064, 874289299456, 69013504
Offset: 1
Keywords
Examples
F(12)=a(1)*a(2)*a(3)*a(4)*a(6)*a(12)=2*1*40*28*16*352=12615680 F(9)=a(2)*a(6)*a(18)= 1*16*8128=130048 L(12)=a(8)*a(24)=496*183808=91168768 L(21)=a(1)*a(3)*a(7)*a(21)=2*40*21568*49205248=84900703109120
Programs
-
Maple
with(numtheory): a[1]:=2:a[2]:=1:for n from 3 to 60 do a[n]:=round(evalf((sqrt(13)-1)^degree(cyclotomic(n, x), x) *cyclotomic(n, (7+sqrt(13))/6), 30)) od: seq(a[n], n=1..60);
Formula
a(n)= (sqrt(13)-1)^degree(cyclotomic(n,x),x)*cyclotomic(n,(7+sqrt(13)/6) L(n)=12*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);
Comments