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A192913 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).

%I A192913 #23 Aug 04 2025 14:47:59
%S A192913 0,0,2,3,10,32,91,273,816,2420,7209,21456,63842,190008,565470,1682835,
%T A192913 5008192,14904512,44356229,132005445,392851940,1169138532,3479389655,
%U A192913 10354762656,30816068600,91709498068,272930078466,812247687927
%N A192913 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 A192913 (See A192911.)
%H A192913 G. C. Greubel, <a href="/A192913/b192913.txt">Table of n, a(n) for n = 0..1000</a>
%H A192913 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,5,2,-1,1).
%F A192913 (See A192911.)
%F A192913 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
%F A192913 a(n) = a(n-1) + 4*a(n-2) + 5*a(n-3) + 2*a(n-4) - a(n-5) + a(n-6). - _Wesley Ivan Hurt_, Aug 04 2025
%e A192913 (See A192911.)
%t A192913 (See A192911.)
%t A192913 LinearRecurrence[{1,4,5,2,-1,1},{0,0,2,3,10,32},28] (* _Ray Chandler_, Aug 02 2015 *)
%o A192913 (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 A192913 (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 A192913 (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 A192913 (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 A192913 Cf. A192232, A192744, A192911.
%K A192913 nonn
%O A192913 0,3
%A A192913 _Clark Kimberling_, Jul 12 2011