%I #27 Oct 17 2018 18:49:49
%S 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,
%T -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,
%U -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
%N A Chebyshev transform of Fib(n+1).
%C 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)).
%C With offset 1 this is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = y = z = 1.
%H Dror Bar-Natan, <a href="http://katlas.org/">The Rolfsen Knot Table</a>
%H C. Kimberling, <a href="http://www.fq.math.ca/Scanned/17-1/kimberling1.pdf">Strong divisibility sequences and some conjectures</a>, Fib. Quart., 17 (1979), 13-17.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1,-1).
%F G.f.: (1+x^2)/(1-x+x^2-x^3+x^4).
%F 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).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*Fib(n-2k+1).
%F a(n) = Sum_{k=0..n} binomial((n+k)/2, k)*(-1)^((n-k)/2)*(1+(-1)^(n+k))*Fib(k+1)/2.
%F a(n) = Sum_{k=0..n} A014019(n-k)*binomial(1, k/2)(1+(-1)^k)/2.
%F 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
%F Euler transform of length 10 sequence [1, 0, 0, -1, -1, 0, 0, 0, 0, 1]. - _Michael Somos_, Sep 18 2006
%F G.f.: (1 - x^4) * (1 - x^5) / ((1 - x) * (1 - x^10)). - _Michael Somos_, Sep 18 2006
%F a(n) = -a(n-5) = -a(-2-n) for all n in Z. - _Michael Somos_, Sep 18 2006
%F Hankel transform is 1,0,0,1,0,0,0,... - _Paul Barry_, Jun 24 2008
%F 0 = (a(n) - a(n+1)) * (a(n) - a(n+1) + a(n+2)) for all n in Z. - _Michael Somos_, Jul 07 2014
%F 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
%F 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
%F a(n) = (-1)^n * A156174(n). - _Michael Somos_, Oct 17 2018
%e G.f. = 1 + x + x^2 + x^3 - x^5 - x^6 - x^7 - x^8 + x^10 + x^11 + x^12 + ...
%t a[ n_] := With[ {m = n + 1}, Sign[Mod[m, 5]] (-1)^Quotient[m, 5]]; (* _Michael Somos_, Jun 17 2015 *)
%t LinearRecurrence[{1, -1, 1, -1}, {1, 1, 1, 1}, 50] (* _G. C. Greubel_, Aug 08 2017 *)
%o (PARI) {a(n) = n++; sign(n%5) * (-1)^(n\5)}; /* _Michael Somos_, Sep 18 2006 */
%Y Cf. A156174.
%K easy,sign
%O 0,1
%A _Paul Barry_, Oct 16 2004