A099461 An Alexander sequence for the knot 9_48.

1, 7, 38, 196, 1001, 5110, 26093, 133252, 680510, 3475339, 17748434, 90640627, 462898478, 2364006148, 12072895733, 61655851222, 314874250049, 1608051650884, 8212262868470, 41939735818687, 214184746483778, 1093833919809295, 5586171115205846, 28528378178106436, 145693417671662033, 744051127629095062, 3799842775146922277, 19405662567631938052, 99104031922539424718

Offset: 0

Author

Paul Barry, Oct 16 2004

Keywords

• easy
• nonn

Comments

The denominator 1 -7*x +11*x^2 -7*x^3 +x^4 is a parameterization of the Alexander polynomial for the knot 9_48. 1/(1 -7*x +11*x^2 -7*x^3 +x^4) is the image of the g.f. of A099459 under the modified Chebyshev transform A(x) -> (1/(1+x^2)^2)*A(x/(1+x^2)).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Dror Bar-Natan, 9 48, The Knot Atlas.
• Index entries for linear recurrences with constant coefficients, signature (7,-11,7,-1).

Crossrefs

Cf. A099459, A099460.

Programs

Magma

I:=[7,38,196,1001]; [1] cat [n le 4 select I[n] else 7*Self(n-1) - 11*Self(n-2) +7*Self(n-3) -Self(n-4): n in [1..41]]; // G. C. Greubel, Nov 18 2021

Mathematica

LinearRecurrence[{7,-11,7,-1},{1,7,38,196,1001},40] (* Harvey P. Dale, Jun 18 2021 *)

Sage

def A099461_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)*(1+x)*(1+x^2)/(1-7*x+11*x^2-7*x^3+x^4) ).list()
A099461_list(40) # G. C. Greubel, Nov 18 2021

Formula

a(n) = A099460(n) - A099460(n-2).

G.f.: (1-x)*(1+x)*(1+x^2)/(1-7*x+11*x^2-7*x^3+x^4). - Corrected by R. J. Mathar, Nov 23 2012