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A192912 Coefficient of x in the reduction by (x^3 -> x + 1) of the polynomial F(n+1)*x^n, where F(n)=A000045 (Fibonacci sequence).

%I A192912 #19 Sep 08 2022 08:45:58
%S A192912 0,1,0,3,10,24,78,231,680,2035,6052,18000,53590,159471,474580,1412397,
%T A192912 4203304,12509144,37227624,110790405,329715412,981242533,2920205614,
%U A192912 8690615136,25863518300,76970566973,229066599960,681708726543
%N A192912 Coefficient of x in the reduction by (x^3 -> x + 1) of the polynomial F(n+1)*x^n, where F(n)=A000045 (Fibonacci sequence).
%C A192912 See A192911.
%H A192912 G. C. Greubel, <a href="/A192912/b192912.txt">Table of n, a(n) for n = 0..1000</a>
%H A192912 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,5,2,-1,1).
%F A192912 (See A192911.)
%F A192912 G.f.: x*(1-x-x^2+2*x^3)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6). - _R. J. Mathar_, May 08 2014
%e A192912 (See A192911.)
%t A192912 (See A192911.)
%t A192912 LinearRecurrence[{1,4,5,2,-1,1},{0,1,0,3,10,24},28] (* _Ray Chandler_, Aug 02 2015 *)
%o A192912 (PARI) my(x='x+O('x^30)); concat([0], Vec(x*(1-x-x^2+2*x^3)/(1-x-4*x^2 -5*x^3-2*x^4+x^5-x^6))) \\ _G. C. Greubel_, Jan 12 2019
%o A192912 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1-x-x^2+2*x^3)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6) )); // _G. C. Greubel_, Jan 12 2019
%o A192912 (Sage) (x*(1-x-x^2+2*x^3)/(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 A192912 (GAP) a:=[0,1,0,3,10,24];; 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 A192912 Cf. A192232, A192744, A192911.
%K A192912 nonn
%O A192912 0,4
%A A192912 _Clark Kimberling_, Jul 12 2011