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A114285 Expansion of g.f. (1 - 3*x)/((1 - x)*(1 - x^2)).

%I A114285 #29 May 16 2025 09:18:06
%S A114285 1,-2,-1,-4,-3,-6,-5,-8,-7,-10,-9,-12,-11,-14,-13,-16,-15,-18,-17,-20,
%T A114285 -19,-22,-21,-24,-23,-26,-25,-28,-27,-30,-29,-32,-31,-34,-33,-36,-35,
%U A114285 -38,-37,-40,-39,-42,-41,-44,-43,-46,-45,-48,-47,-50,-49,-52,-51,-54,-53,-56,-55,-58,-57,-60,-59,-62,-61
%N A114285 Expansion of g.f. (1 - 3*x)/((1 - x)*(1 - x^2)).
%C A114285 Diagonal sums of A114284.
%H A114285 Bérénice Delcroix-Oger and Clément Dupont, <a href="https://arxiv.org/abs/2505.06094">Lie-operads and operadic modules from poset cohomology</a>, arXiv:2505.06094 [math.CO], 2025. See p. 28.
%H A114285 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A114285 a(n) = Sum_{k=0..floor(n/2)} 3*0^(n-2k) - 2.
%F A114285 a(n) = 3*(1 + (-1)^n)/2 - 2*floor((n+2)/2).
%F A114285 a(n) = - A103889(n). - _R. J. Mathar_, Apr 06 2008
%F A114285 From _Wesley Ivan Hurt_, Sep 06 2015: (Start)
%F A114285 a(n) = a(n-1) + a(n-2) - a(n-3), n > 2.
%F A114285 a(n) = (-1)^n - n. (End)
%F A114285 E.g.f.: exp(-x) - x*exp(x). - _Stefano Spezia_, May 03 2023
%p A114285 A114285:=n->(-1)^n-n: seq(A114285(n), n=0..70); # _Wesley Ivan Hurt_, Sep 06 2015
%t A114285 Table[(-1)^n-n, {n,0,70}] (* _Wesley Ivan Hurt_, Sep 06 2015 *)
%t A114285 CoefficientList[Series[(1 - 3 x)/((1 - x) (1 - x^2)), {x, 0, 70}] ,x] (* _Vincenzo Librandi_, Sep 07 2015 *)
%t A114285 LinearRecurrence[{1,1,-1},{1,-2,-1},70] (* _Harvey P. Dale_, Jul 24 2019 *)
%o A114285 (Magma) [(-1)^n-n : n in [0..70]]; // _Wesley Ivan Hurt_, Sep 06 2015
%o A114285 (Magma) I:=[1,-2,-1]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..70]]; // _Vincenzo Librandi_, Sep 07 2015
%Y A114285 Cf. A103889, A114284.
%K A114285 easy,sign
%O A114285 0,2
%A A114285 _Paul Barry_, Nov 20 2005