A099860 A Chebyshev transform related to the knot 7_1.

1, 1, 2, 2, 1, 1, 0, -1, -1, -2, -2, -1, -1, 0, 1, 1, 2, 2, 1, 1, 0, -1, -1, -2, -2, -1, -1, 0, 1, 1, 2, 2, 1, 1, 0, -1, -1, -2, -2, -1, -1, 0, 1, 1, 2, 2, 1, 1, 0, -1, -1, -2, -2, -1, -1, 0, 1, 1, 2, 2, 1, 1, 0, -1, -1, -2, -2, -1, -1, 0, 1, 1, 2, 2, 1, 1, 0, -1, -1, -2, -2

Offset: 0

Author

Paul Barry, Oct 28 2004

Keywords

• easy
• sign

Comments

The g.f. is the transform of the g.f. of A006053(n+1) under the Chebyshev mapping G(x)-> (1/(1+x^2))G(x/(1+x^2)). The denominator of the g.f. is a parameterization of the Alexander polynomial of 7_1. It is also the 14th cyclotomic polynomial.

Links

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

Crossrefs

Cf. A099859.

Programs

Mathematica

LinearRecurrence[{1,-1,1,-1,1,-1},{1,1,2,2,1,1},100] (* Harvey P. Dale, May 21 2019 *)

Formula

G.f.: (1+x^2)^2/(1-x+x^2-x^3+x^4-x^5+x^6); a(n)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^k*A006053(n-2k+1)}.