A099846 An Alexander sequence for the knot 8_5.

1, 3, 5, 8, 15, 29, 55, 104, 196, 368, 692, 1304, 2457, 4627, 8713, 16408, 30899, 58189, 109583, 206368, 388632, 731872, 1378264, 2595552, 4887953, 9205011, 17334909, 32645160, 61477479, 115774605, 218027143, 410589480, 773223548, 1456137296

Offset: 0

Author

Paul Barry, Oct 27 2004

Keywords

• easy
• nonn

Comments

The g.f. is a transformation of the g.f. 1/((1-x)(1-2x-x^2)) of A048739 under the mapping G(x)->(1/(1+x^2)^3)G(x/(1+x^2)). The denominator of the g.f. is a parameterization of the Alexander polynomial of the knot 8_5. Relates 8_5 to the Pell numbers.

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-4,5,-4,3,-1).

Crossrefs

Cf. A099854.

Programs

Mathematica

CoefficientList[Series[1/(1-3x+4x^2-5x^3+4x^4-3x^5+x^6),{x,0,40}],x] (* or *) LinearRecurrence[{3,-4,5,-4,3,-1},{1,3,5,8,15,29},41] (* Harvey P. Dale, Sep 25 2011 *)

Formula

G.f.: 1/(1-3x+4x^2-5x^3+4x^4-3x^5+x^6).

a(0)=1, a(1)=3, a(2)=5, a(3)=8, a(4)=15, a(5)=29, a(n)=3*a(n-1)- 4*a(n-2)+ 5*a(n-3)-4*a(n-4)+3*a(n-5)-a(n-6). - Harvey P. Dale, Sep 25 2011