%I #32 Mar 12 2021 22:24:43

%S 1,1,-1,-2,-1,1,2,1,-1,-2,-1,1,2,1,-1,-2,-1,1,2,1,-1,-2,-1,1,2,1,-1,

%T -2,-1,1,2,1,-1,-2,-1,1,2,1,-1,-2,-1,1,2,1,-1,-2,-1,1,2,1,-1,-2,-1,1,

%U 2,1,-1,-2,-1,1,2,1,-1,-2,-1,1,2,1,-1,-2,-1,1,2,1,-1

%N A Chebyshev transform of 1,1,1,...

%C A Chebyshev transform of 1/(1-x): if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).

%C Transform of 1/(1+x) under the mapping g(x)->((1+x)/(1-x))g(x/(1-x)^2). - _Paul Barry_, Dec 01 2004

%C Multiplicative with a(p^e) = -1 if p = 2; -2 if p = 3; 1 otherwise. - _David W. Wilson_ Jun 10 2005

%H G. C. Greubel, <a href="/A100051/b100051.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/rfmc.txt">Rational Function Multiplicative Coefficients</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1).

%F From _Paul Barry_, Dec 01 2004: (Start)

%F G.f.: (1-x^2)/(1-x+x^2).

%F a(n) = a(n-1) - a(n-2), n>2.

%F a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)/(n-k).

%F a(n) = Sum_{k=0..n} binomial(n+k, 2k)*(2n/(n+k))*(-1)^k, n>1. (End)

%F Moebius transform is length 6 sequence [1, -2, -3, 0, 0, 6].

%F Euler transform of length 6 sequence [1, -2, -1, 0, 0, 1].

%F a(n) = a(-n). a(n) = c_6(n) if n>1 where c_k(n) is Ramanujan's sum. - _Michael Somos_, Mar 21 2011

%F a(n) = A087204(n), n>0. - _R. J. Mathar_, Sep 02 2008

%F a(n) = A057079(n+1), n>0. Dirichlet g.f. zeta(s) *(1-2^(1-s)-3^(1-s)+6^(1-s)). - _R. J. Mathar_, Apr 11 2011

%e G.f. = 1 + x - x^2 - 2*x^3 - x^4 + x^5 + 2*x^6 + x^7 - x^8 - 2*x^9 - x^10 + ...

%t CoefficientList[Series[(1 - x^2)/(1 - x + x^2), {x,0,50}], x] (* _G. C. Greubel_, May 03 2017 *)

%t LinearRecurrence[{1,-1},{1,1,-1},80] (* _Harvey P. Dale_, Mar 25 2019 *)

%o (PARI) {a(n) = - (n == 0) + [2, 1, -1, -2, -1, 1][n%6 + 1]}; /* _Michael Somos_, Mar 21 2011 */

%Y Cf. A099837, A099443, A011655, A100047, A100048, A100050.

%Y Row sums of array A127677.

%K easy,sign,mult

%O 0,4

%A _Paul Barry_, Oct 31 2004