%I #24 Sep 08 2022 08:45:43
%S 1,0,0,1,-1,-1,1,-2,-4,1,-3,-9,1,-4,-16,1,-5,-25,1,-6,-36,1,-7,-49,1,
%T -8,-64,1,-9,-81,1,-10,-100,1,-11,-121,1,-12,-144,1,-13,-169,1,-14,
%U -196,1,-15,-225,1,-16,-256,1,-17,-289,1,-18,-324,1,-19,-361,1,-20,-400,1,-21,-441
%N Expansion of (1 - 2*x^3 - x^4 - x^5 + x^6 + x^7 - x^8)/(1 - x^3)^3.
%C The sequence is given by the successive triples (1, -m, -m^2) with m = 0, 1, 2, 3, ... - _Bruno Berselli_, Aug 23 2018
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,3,0,0,-3,0,0,1).
%F From _Bruno Berselli_, Aug 23 2018: (Start)
%F G.f.: (1 - 2*x^3 - x^4 - x^5 + x^6 + x^7 - x^8)/((1 - x)^3*(1 + x + x^2)^3).
%F a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) for n>8.
%F a(n) = -(-1)^((n+1) mod 3)*floor(n/3)^(n mod 3). (End)
%e As array:
%e 1, 0, 0;
%e 1, -1, -1;
%e 1, -2, -4;
%e 1, -3, -9;
%e 1, -4, -16;
%e 1, -5, -25;
%e 1, -6, -36;
%e 1, -7, -49;
%e 1, -8, -64;
%e 1, -9, -81;
%e 1, -10, -100 etc.
%t CoefficientList[Series[(1-2x^3-x^4-x^5+x^6+x^7-x^8)/(1-x^3)^3,{x,0,100}],x] (* or *) LinearRecurrence[{0,0,3,0,0,-3,0,0,1},{1,0,0,1,-1,-1,1,-2,-4},100] (* _Harvey P. Dale_, Nov 22 2021 *)
%o (Magma) &cat [[1,-n,-n^2]: n in [0..25]]; // _Bruno Berselli_, Aug 23 2018
%Y Cf. A000463, A156133, A234357.
%K sign,tabf,easy,less
%O 0,8
%A _Roger L. Bagula_, Mar 22 2009
%E Edited, new name, and a(1)-a(2) added by _Bruno Berselli_, Aug 23 2018