A099455 An Alexander sequence for the knot 8_12.

1, 7, 36, 168, 755, 3346, 14747, 64848, 284892, 1251103, 5493314, 24118255, 105887532, 464877504, 2040939083, 8960260498, 39337870403, 172703402424, 758212386132, 3328747303735, 14614056052994, 64159460722903, 281676515111412, 1236632261449368, 5429133302704547

Offset: 0

Author

Paul Barry, Oct 16 2004

Keywords

• easy
• nonn

Comments

The denominator is a parameterization of the Alexander polynomial for the knot 8_12. 1/(1-7*x+13*x^2-7*x^3+x^4) is the image of the g.f. of A099453 under the modified Chebyshev transform A(x)->(1/(1+x^2)^2)A(x/(1+x^2)).

Links

• Stefano Spezia, Table of n, a(n) for n = 0..1500
• Dror Bar-Natan, The Rolfsen Knot Table.
• Index entries for linear recurrences with constant coefficients, signature (7,-13,7,-1).

Crossrefs

Cf. A099454.

Programs

Mathematica

LinearRecurrence[{7,-13,7,-1},{1,7,36,168,755},30] (* Harvey P. Dale, Jan 31 2017 *)

Formula

G.f.: (1-x)*(1+x)*(1+x^2)/(1-7*x+13*x^2-7*x^3+x^4). - corrected Nov 24 2012

a(n) = A099454(n) - A099454(n-2).