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%I #12 Sep 08 2022 08:46:06
%S 1,0,-1,-1,1,4,4,-3,-14,-15,9,49,56,-26,-171,-208,71,595,769,-176,
%T -2064,-2831,354,7137,10381,-295,-24596,-37926,-2359,84464,138079,
%U 20407,-288959,-501060,-114836,984549,1812546,556609,-3339871,-6537023,-2497824,11275550
%N a(n) = 2*a(n-1) - 3*a(n-2) + a(n-3), a(0) = 1, a(1) = 0, a(2) = -1.
%H G. C. Greubel, <a href="/A233581/b233581.txt">Table of n, a(n) for n = 0..2500</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,-3,1).
%F G.f.: (1 - 2*x + 2*x^2) / (1 - 2*x + 3*x^2 - x^3).
%F a(n) = A052921(-n). a(n)^2 - a(n-1)*a(n+1) = A034943(n).
%F a(n) = A127896(n) -2*A127896(n-1) + 2*A127896(n-2). - _R. J. Mathar_, Sep 24 2021
%e G.f. = 1 - x^2 - x^3 + x^4 + 4*x^5 + 4*x^6 - 3*x^7 - 14*x^8 - 15*x^9 + ...
%t CoefficientList[Series[(1-2*x+2*x^2)/(1-2*x+3*x^2-x^3), {x, 0, 50}], x] (* or *) LinearRecurrence[{2,-3,1}, {1,0,-1}, 50] (* _G. C. Greubel_, Aug 08 2018 *)
%o (PARI) {a(n) = if( n<0, polcoeff( (1 - x) / (1 - 3*x + 2*x^2 - x^3) + x * O(x^-n), -n), polcoeff( (1 - 2*x + 2*x^2) / (1 - 2*x + 3*x^2 - x^3) + x * O(x^n), n))}
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x+2*x^2)/(1-2*x+3*x^2-x^3))); // _G. C. Greubel_, Aug 08 2018
%Y Cf. A034943, A052921.
%K sign
%O 0,6
%A _Michael Somos_, Dec 14 2013
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