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A280166 a(2*n) = 4*n if n>0, a(2*n + 1) = -(2*n + 1), a(0) = 1.

%I A280166 #32 Sep 08 2022 08:46:18
%S A280166 1,-1,4,-3,8,-5,12,-7,16,-9,20,-11,24,-13,28,-15,32,-17,36,-19,40,-21,
%T A280166 44,-23,48,-25,52,-27,56,-29,60,-31,64,-33,68,-35,72,-37,76,-39,80,
%U A280166 -41,84,-43,88,-45,92,-47,96,-49,100,-51,104,-53,108,-55,112,-57,116
%N A280166 a(2*n) = 4*n if n>0, a(2*n + 1) = -(2*n + 1), a(0) = 1.
%H A280166 G. C. Greubel, <a href="/A280166/b280166.txt">Table of n, a(n) for n = 0..5000</a>
%H A280166 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A280166 Euler transform of length 6 sequence [-1, 4, 1, -1, 0, -1].
%F A280166 a(n) = (-1)^n * A257088(n), with A257088 multiplicative (see there).
%F A280166 a(n) = n * A168361(n+1) if n>0.
%F A280166 a(2*n) = A008574(n). a(2*n + 1) = - A005408(n).
%F A280166 G.f.: (1 - x + x^2) * (1 + x^2) / (1 - x^2)^2.
%e A280166 G.f. = 1 - x + 4*x^2 - 3*x^3 + 8*x^4 - 5*x^5 + 12*x^6 - 7*x^7 + 16*x^8 + ...
%t A280166 a[ n_] := Which[ n < 1, Boole[n == 0], OddQ[n], -n, True, 2 n];
%t A280166 a[ n_] := SeriesCoefficient[ (1 - x + x^2) (1 + x^2) / (1 - x^2)^2, {x, 0, n}];
%o A280166 (PARI) {a(n) = if( n<1, n==0, n%2, -n, 2*n)};
%o A280166 (PARI) x='x+O('x^50); Vec((1-x+x^2)*(1+x^2)/(1-x^2)^2) \\ _G. C. Greubel_, Aug 04 2018
%o A280166 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 - x+x^2)*(1+x^2)/(1-x^2)^2)); // _G. C. Greubel_, Aug 04 2018
%Y A280166 Cf. A005408, A008574, A168361, A257088.
%K A280166 sign,easy
%O A280166 0,3
%A A280166 _Michael Somos_, Dec 27 2016
%E A280166 Edited by _M. F. Hasler_, May 08 2018