A099445 An Alexander sequence for the Miller Institute knot.

1, 3, 6, 12, 25, 54, 117, 252, 542, 1167, 2514, 5415, 11662, 25116, 54093, 116502, 250913, 540396, 1163862, 2506635, 5398594, 11627067, 25041462, 53932332, 116155217, 250165974, 538787805, 1160398812, 2499175726, 5382528183

Offset

0,2

Comments

The denominator is a parameterization of the Alexander polynomial for the knot 6_2 (Miller Institute knot). The g.f. is the image of that of Fib(2n+2) under the modified Chebyshev transform A(x)->(1/(1+x^2)^2)A(x/(1+x^2)).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Dror Bar-Natan, The Rolfsen Knot Table

Index entries for linear recurrences with constant coefficients, signature (3,-3,3,-1).

Formula

G.f.: -(x-1)*(x+1)*(x^2+1) / (x^4-3*x^3+3*x^2-3*x+1). - Colin Barker, Feb 10 2014

a(n) = A099444(n)-A099444(n-2).

a(n) = 3*a(n-1)-3*a(n-2)+3*a(n-3)-a(n-4) for n>4. - Colin Barker, Feb 10 2014

Mathematica

CoefficientList[Series[(1 - x) (x + 1) (x^2 + 1)/(x^4 - 3 x^3 + 3 x^2 - 3 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{3, -3, 3, -1}, {1, 3, 6, 12, 25}, 30] (* Harvey P. Dale, Jun 24 2018 *)

Prog

(PARI) Vec(-(x-1)*(x+1)*(x^2+1)/(x^4-3*x^3+3*x^2-3*x+1) + O(x^100)) \\ Colin Barker, Feb 10 2014
(Magma) I:=[1, 3, 6, 12, 25, 54, 117, 252]; [n le 8 select I[n] else 3*Self(n-1)-3*Self(n-2)+3*Self(n-3)-Self(n-4) : n in [1..50]]; // Vincenzo Librandi, Feb 12 2014

Crossrefs

Cf. A001906.

Sequence in context: A243728 A243721 A280109 * A004067 A278821 A274476

Adjacent sequences: A099442 A099443 A099444 * A099446 A099447 A099448

Keyword

easy,nonn

Author

Paul Barry, Oct 16 2004

Extensions

g.f. corrected by Colin Barker, Feb 10 2014

Status

approved