A099452 An Alexander sequence for the knot 7_7.

1, 5, 16, 40, 79, 110, 23, -520, -2336, -6995, -16574, -31075, -38848, 9560, 258631, 1043950, 2978719, 6781640, 12060848, 13119125, -12022526, -124662155, -461573264, -1259138680, -2752822273, -4615067410, -4134056729, 8360350360, 58685747584, 202130368445, 528415922498

Offset: 0

Author

Paul Barry, Oct 16 2004

Keywords

• easy
• sign

Comments

The denominator is a parameterization of the Alexander polynomial for the knot 7_7. 1/(1-5*x+9*x^2-5*x^3+x^4) is the image of the g.f. of A099450 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 (5,-9,5,-1).

Programs

Mathematica

LinearRecurrence[{5,-9,5,-1},{1,5,16,40,79},40] (* Harvey P. Dale, Apr 18 2019 *)

Formula

G.f.: (1-x)*(1+x)*(1+x^2)/(1-5*x+9*x^2-5*x^3+x^4). - corrected by R. J. Mathar, Nov 24 2012

a(n)=A099451(n)-A099451(n-2).