A099447 An Alexander sequence for the knot 6_3.

1, 3, 4, 0, -13, -30, -29, 24, 140, 243, 130, -429, -1348, -1752, 67, 5346, 11795, 10608, -11180, -56541, -93694, -42525, 182452, 535440, 660179, -106782, -2197373, -4613112, -3832996, 5081235, 22766722, 36008115

Offset: 0

Author

Paul Barry, Oct 16 2004

Keywords

• easy
• sign

Comments

The denominator is a parameterization of the Alexander polynomial for the knot 6_3. 1/(1-3*x+5*x^2-3*x^3+x^4) is the image of the g.f. of A057083 under the modified Chebyshev transform A(x)->(1/(1+x^2)^2)A(x/(1+x^2)).

Links

• Dror Bar-Natan, The Rolfsen Knot Table
• Index entries for linear recurrences with constant coefficients, signature (3,-5,3,-1).

Programs

Mathematica

LinearRecurrence[{3,-5,3,-1},{1,3,4,0,-13},40] (* Harvey P. Dale, Oct 07 2017 *)

Formula

G.f.: (1-x)*(1+x)*(1+x^2)/(1-3x+5x^2-3x^3+x^4); - corrected Nov 24 2012

a(n)=A099446(n)-A099446(n-2).