A099443 A Chebyshev transform of Fib(n+1).

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

Offset

0,1

Comments

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)).

With offset 1 this is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = y = z = 1.

Links

Table of n, a(n) for n=0..104.

Dror Bar-Natan, The Rolfsen Knot Table

C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.

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

Formula

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

a(n) = sqrt(2*(10 - 4*sqrt(5))/25)*cos((6*Pi*n +Pi)/10) + sqrt(2*(4*sqrt(5) + 10)/25)*sin(Pi*(n+1)/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_, 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)). - _Michael Somos_, Sep 18 2006

a(n) = -a(n-5) = -a(-2-n) for all n in Z. - _Michael Somos_, Sep 18 2006

Hankel transform is 1,0,0,1,0,0,0,... - _Paul Barry_, Jun 24 2008

0 = (a(n) - a(n+1)) * (a(n) - a(n+1) + a(n+2)) for all n in Z. - _Michael Somos_, Jul 07 2014

0 = a(n)*a(n-4) - a(n-1)*a(n-3) + a(n-2)*a(n-2) for all n in Z. - _Michael Somos_, Jul 07 2014

0 = a(n)*a(n+5) - a(n+1)*a(n+4) + a(n+2)*a(n+3) for all n in Z. - _Michael Somos_, Jul 07 2014

a(n) = (-1)^n * A156174(n). - _Michael Somos_, Oct 17 2018

Example

G.f. = 1 + x + x^2 + x^3 - x^5 - x^6 - x^7 - x^8 + x^10 + x^11 + x^12 + ...

Mathematica

a[ n_] := With[ {m = n + 1}, Sign[Mod[m, 5]] (-1)^Quotient[m, 5]]; (* _Michael Somos_, Jun 17 2015 *)
LinearRecurrence[{1, -1, 1, -1}, {1, 1, 1, 1}, 50] (* _G. C. Greubel_, Aug 08 2017 *)

Prog

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

Crossrefs

Cf. A156174.

Sequence in context: A190239 A120529 A292301 * A132342 A156174 A358847

Adjacent sequences: A099440 A099441 A099442 * A099444 A099445 A099446

Keyword

easy,sign

Author

_Paul Barry_, Oct 16 2004

Status

approved