OFFSET
0,2
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).
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}.
MATHEMATICA
LinearRecurrence[{3, -3, 3, -1}, {1, 3, 7, 15}, 30] (* Harvey P. Dale, Sep 30 2018 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Oct 16 2004
STATUS
approved