A099444 A Chebyshev transform of Fib(2n+2).

1, 3, 7, 15, 32, 69, 149, 321, 691, 1488, 3205, 6903, 14867, 32019, 68960, 148521, 319873, 688917, 1483735, 3195552, 6882329, 14822619, 31923791, 68754951, 148079008, 318920925, 686866813, 1479319737, 3186042539, 6861847920

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 the g.f. of Fib(2n+2) under the Chebyshev transform A(x)->(1/(1+x^2))A(x/(1+x^2)).

This sequence is the p-INVERT of A010892 using p(S) = 1 - S - S^2; see A292324. - Clark Kimberling, Sep 26 2017

Links

• Harvey P. Dale, 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

Mathematica

LinearRecurrence[{3,-3,3,-1},{1,3,7,15},30] (* Harvey P. Dale, Sep 30 2018 *)

Formula

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

a(n) = sum{k=0..floor(n/2), binomial(n-k, k)(-1)^k*Fib(2(n-2k)+2)};

a(n) = sum{k=0..n, binomial((n+k)/2, k)(-1)^((n-k)/2)(1+(-1)^(n+k))Fib(2k+2)/2};

a(n) = sum{k=0..n, A099445(n-k)*binomial(1, k/2)(1+(-1)^k)/2}.