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

Author

Paul Barry, Oct 16 2004

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A001906.

Programs

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

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 *)

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

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

Extensions

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