A099449 An Alexander sequence for the knot 7_6.

1, 5, 18, 60, 197, 650, 2153, 7140, 23682, 78545, 260498, 863945, 2865282, 9502740, 31515953, 104523050, 346651997, 1149675660, 3812913618, 12645575405, 41939208002, 139091904605, 461300030418, 1529907284460, 5073956524397

Offset: 0

Author

Paul Barry, Oct 16 2004

Keywords

• easy
• nonn

Comments

The denominator is a parameterization of the Alexander polynomial for the knot 7_6. The g.f. is the image of the g.f. of A030191 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
• The Rolfsen Knot Table
• Index entries for linear recurrences with constant coefficients, signature (5,-7,5,-1).

Crossrefs

Cf. A030191, A099448.

Programs

Magma

I:=[1,5,18,60,197,650,2153,7140]; [n le 8 select I[n] else 5*Self(n-1)-7*Self(n-2)+5*Self(n-3)-Self(n-4) : n in [1..30]]; // Vincenzo Librandi, Feb 12 2014

Mathematica

CoefficientList[Series[(1 - x) (x + 1) (x^2 + 1)/(x^4 -5 x^3 + 7 x^2 - 5 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{5,-7,5,-1},{1,5,18,60,197},30] (* Harvey P. Dale, Oct 06 2015 *)

PARI

Vec(-(x-1)*(x+1)*(x^2+1)/(x^4-5*x^3+7*x^2-5*x+1) + O(x^100)) \\ Colin Barker, Feb 10 2014

Formula

G.f.: -(x-1)*(x+1)*(x^2+1) / (x^4-5*x^3+7*x^2-5*x+1). - Colin Barker, Feb 10 2014

a(n) = A099448(n) - A099448(n-2).

a(n) = 5*a(n-1)-7*a(n-2)+5*a(n-3)-a(n-4) for n>4. - Colin Barker, Feb 10 2014

Extensions

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