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A192807 Coefficient of x in the reduction of the polynomial (x^2 + x + 1)^n by x^3 -> x^2 + x + 1.

%I A192807 #19 Sep 08 2022 08:45:58
%S A192807 0,1,6,37,230,1431,8904,55403,344732,2145013,13346834,83047505,
%T A192807 516743378,3215312955,20006521300,124485827703,774583500376,
%U A192807 4819661885417,29989201523742,186600684739485,1161078447443102,7224534909928031
%N A192807 Coefficient of x in the reduction of the polynomial (x^2 + x + 1)^n by x^3 -> x^2 + x + 1.
%C A192807 For discussions of polynomial reduction, see A192232 and A192744.
%H A192807 G. C. Greubel, <a href="/A192807/b192807.txt">Table of n, a(n) for n = 0..1000</a>
%H A192807 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-5,1).
%F A192807 a(n) = 7*a(n-1) - 5*a(n-2) + a(n-3).
%F A192807 G.f.: x*(x - 1)/(x^3 - 5*x^2 + 7*x - 1). - _Colin Barker_, Nov 23 2012
%t A192807 (See A192806.)
%t A192807 LinearRecurrence[{7,-5,1},{0,1,6},30] (* _Harvey P. Dale_, Oct 09 2017 *)
%o A192807 (PARI) my(x='x+O('x^30)); concat([0], Vec(x*(x-1)/(x^3-5*x^2+7*x-1))) \\ _G. C. Greubel_, Jan 02 2019
%o A192807 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(x-1)/(x^3-5*x^2+7*x-1) )); // _G. C. Greubel_, Jan 02 2019
%o A192807 (Sage) (x*(x-1)/(x^3-5*x^2+7*x-1)).series(x,30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 02 2019
%o A192807 (GAP) a:=[0,1,6];; for n in [4..25] do a[n]:=7*a[n-1]-5*a[n-2]+a[n-3]; od; Print(a); # _Muniru A Asiru_, Jan 02 2019
%Y A192807 Cf. A192744, A192232, A192806, A099464.
%K A192807 nonn,easy
%O A192807 0,3
%A A192807 _Clark Kimberling_, Jul 10 2011