0,1

The denominator is the 10th cyclotomic polynomial. It is also associated to the knots 4_1 and 5_1 by the Alexander and Jones polynomials respectively. The g.f. is the image of the g.f. of Fib(n+1) under the Chebyshev transform A(x)->(1/(1+x^2))A(x/(1+x^2)).

The Rolfsen Knot Table

G.f.: (1+x^2)/(1-x+x^2-x^3+x^4) a(n)=sqrt(4/5-8sqrt(5)/25)cos(3*pi*n/5+pi/10)+sqrt(8sqrt(5)/25+4/5)sin(pi*n/5+pi/5); a(n)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^k*Fib(n-2k+1)}; a(n)=sum{k=0..n, binomial((n+k)/2, k)(-1)^((n-k)/2)(1+(-1)^(n+k))Fib(k+1)/2}; a(n)=sum{k=0..n, A014019(n-k)*binomial(1, k/2)(1+(-1)^k)/2}.

With a leading zero, this is sum{k=0..floor(n/2), binomial(n-k-1, k)(-1)^kFib(n-2k)}, or the image of x/(1-x-x^2) under the mapping g(x)->g(x/(1+x^2)). - Paul Barry (pbarry(AT)wit.ie), Jan 16 2005

Euler transform of length 10 sequence [ 1, 0, 0, -1, -1, 0, 0, 0, 0, 1]. - Michael Somos Sep 18 2006

G.f.: (1-x^4)(1-x^5)/((1-x)(1-x^10)). a(n)=-a(n-5)=-a(-2-n). - Michael Somos Sep 18 2006

Hankel transform is 1,0,0,1,0,0,0,... - Paul Barry (pbarry(AT)wit.ie), Jun 24 2008

(PARI) {a(n)=n++; sign(n%5)*(-1)^(n\5)} /* Michael Somos Sep 18 2006 */

Adjacent sequences: A099440 A099441 A099442 this_sequence A099444 A099445 A099446

Sequence in context: A051731 A135839 A120529 this_sequence A132342 A106467 A106468

easy,sign

Paul Barry (pbarry(AT)wit.ie), Oct 16 2004