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%I #21 Sep 08 2022 08:45:58
%S 0,0,2,3,10,32,91,273,816,2420,7209,21456,63842,190008,565470,1682835,
%T 5008192,14904512,44356229,132005445,392851940,1169138532,3479389655,
%U 10354762656,30816068600,91709498068,272930078466,812247687927
%N Coefficient of x^2 in the reduction by (x^3 -> x + 1) of the polynomial F(n+1)*x^n, where F(n)=A000045 (Fibonacci sequence).
%C (See A192911.)
%H G. C. Greubel, <a href="/A192913/b192913.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,5,2,-1,1).
%F (See A192911.)
%F G.f.: x^2*(1+x)*(2-x) / (1 - x - 4*x^2 - 5*x^3 - 2*x^4 + x^5 - x^6). - _R. J. Mathar_, May 08 2014
%e (See A192911.)
%t (See A192911.)
%t LinearRecurrence[{1,4,5,2,-1,1},{0,0,2,3,10,32},28] (* _Ray Chandler_, Aug 02 2015 *)
%o (PARI) my(x='x+O('x^30)); concat([0,0], Vec(x^2*(1+x)*(2-x)/(1-x-4*x^2 -5*x^3-2*x^4+x^5-x^6))) \\ _G. C. Greubel_, Jan 12 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!( x^2*(1+x)*(2-x)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6) )); // _G. C. Greubel_, Jan 12 2019
%o (Sage) (x^2*(1+x)*(2-x)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 12 2019
%o (GAP) a:=[0,0,2,3,10,32];; for n in [7..30] do a[n]:=a[n-1]+4*a[n-2] +5*a[n-3]+2*a[n-4]-a[n-5]+a[n-6]; od; a; # _G. C. Greubel_, Jan 12 2019
%Y Cf. A192232, A192744, A192911.
%K nonn
%O 0,3
%A _Clark Kimberling_, Jul 12 2011