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A309215 a(0)=0; thereafter a(n) = a(n-1)+n if a(n-1) odd, otherwise a(n) = a(n-1)-n.

%I A309215 #23 Sep 26 2021 14:38:54
%S A309215 0,-1,1,4,0,-5,1,8,0,-9,1,12,0,-13,1,16,0,-17,1,20,0,-21,1,24,0,-25,1,
%T A309215 28,0,-29,1,32,0,-33,1,36,0,-37,1,40,0,-41,1,44,0,-45,1,48,0,-49,1,52,
%U A309215 0,-53,1,56,0,-57,1,60,0,-61,1,64,0,-65,1,68,0,-69,1,72,0,-73,1,76,0,-77
%N A309215 a(0)=0; thereafter a(n) = a(n-1)+n if a(n-1) odd, otherwise a(n) = a(n-1)-n.
%C A309215 A003816 and A309214 have the same terms except for signs.
%H A309215 Colin Barker, <a href="/A309215/b309215.txt">Table of n, a(n) for n = 0..1000</a>
%H A309215 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,-2,2,-1,1).
%F A309215 a(4t)=0, a(4t+1)=-(4t+1), a(4t+2)=1, a(4t+3)=4t+4.
%F A309215 From _Colin Barker_, Aug 13 2019: (Start)
%F A309215 G.f.: -x*(1 - 2*x - x^2) / ((1 - x)*(1 + x^2)^2).
%F A309215 a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-5) for n>4.
%F A309215 a(n) = 1/2 - (1/4 - i/4)*((-i)^n+i^(1+n)) - (1/2)*i*((-i)^n-i^n)*(1+n) where i=sqrt(-1).
%F A309215 (End)
%p A309215 t:=0;
%p A309215 a:=[t]; M:=100;
%p A309215 for i from 1 to M do
%p A309215 if (t mod 2) = 1 then t:=t+i else t:=t-i; fi;
%p A309215 a:=[op(a),t]; od:
%p A309215 a;
%t A309215 nxt[{n_,a_}]:={n+1,If[OddQ[a],a+n+1,a-n-1]}; NestList[nxt,{0,0},80][[All,2]] (* _Harvey P. Dale_, Sep 26 2021 *)
%o A309215 (PARI) concat(0, Vec(-x*(1 - 2*x - x^2) / ((1 - x)*(1 + x^2)^2) + O(x^40))) \\ _Colin Barker_, Aug 13 2019
%Y A309215 Cf. A003816, A309529, A309214, A309216, A309217.
%K A309215 sign,easy
%O A309215 0,4
%A A309215 _N. J. A. Sloane_, Aug 10 2019