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A099461 An Alexander sequence for the knot 9_48. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

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

MATHEMATICA

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

PROG

(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

(Sage)

def A099461_list(prec):

    P.<x> = 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

CROSSREFS

Cf. A099459, A099460.

Sequence in context: A014827 A141845 A048437 * A104553 A027241 A292761

Adjacent sequences:  A099458 A099459 A099460 * A099462 A099463 A099464

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Oct 16 2004

STATUS

approved

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Last modified January 18 11:57 EST 2022. Contains 350455 sequences. (Running on oeis4.)