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A192911 Constant term in the reduction by (x^3 -> x + 1) of the polynomial F(n+1)*x^n, where F(n)=A000045 (Fibonacci sequence).

%I A192911 #19 Sep 08 2022 08:45:58
%S A192911 1,0,0,3,5,16,52,147,442,1320,3916,11664,34717,103298,307440,914949,
%T A192911 2722885,8103424,24116008,71769885,213589298,635647790,1891705884,
%U A192911 5629770720,16754357925,49861446392,148389084968,441610143507
%N A192911 Constant term in the reduction by (x^3 -> x + 1) of the polynomial F(n+1)*x^n, where F(n)=A000045 (Fibonacci sequence).
%C A192911 Regarding polynomial reduction, see A192232 and A192744.  In the case of the reduction at A192911, each term in the three resulting sequences is a product of a Fibonacci number and a tribonacci numbers
%C A192911 A192911(n) = F(n+1)*T3(n+1), where F=A000045, T3=A000073.
%C A192911 A192912(n) = F(n+1)*T2(n), where T2=A001590.
%C A192911 A192913(n) = F(n+1)*T3(n).
%C A192911 All three obey the same linear recurrence, shown below at Formula.
%H A192911 G. C. Greubel, <a href="/A192911/b192911.txt">Table of n, a(n) for n = 0..1000</a>
%H A192911 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,5,2,-1,1).
%F A192911 a(n) = a(n-1) + 4*a(n-2) + 5*a(n-3) + 2*a(n-4) - a(n-5) + a(n-6).
%F A192911 G.f.: (x+1)*(2*x^2+2*x-1)/(x^6-x^5+2*x^4+5*x^3+4*x^2+x-1). - _Colin Barker_, Aug 31 2012
%e A192911 The first six polynomials and reductions:
%e A192911 1 -> 1
%e A192911 x -> 2
%e A192911 2*x^2 -> 2*x^2
%e A192911 3*x^3 -> 3 + 3*x + 3*x^2
%e A192911 5*x^4 -> 5 + 10*x + 10*x^2
%e A192911 8*x^5 -> 16 + 24*x + 32*x^2
%t A192911 q = x^3; s = x^2 + x + 1; z = 22;
%t A192911 p[0, x_] := 1; p[1, x_] := x;
%t A192911 p[n_, x_] := p[n - 1, x]*x + p[n - 2, x]*x^2;
%t A192911 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192911 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192911 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192911 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192911 *)
%t A192911 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192912 *)
%t A192911 u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}] (* A192913 *)
%t A192911 LinearRecurrence[{1,4,5,2,-1,1},{1,0,0,3,5,16},28] (* _Ray Chandler_, Aug 02 2015 *)
%o A192911 (PARI) my(x='x+O('x^30)); Vec((x+1)*(2*x^2+2*x-1)/(x^6-x^5+2*x^4+5*x^3 +4*x^2+x-1)) \\ _G. C. Greubel_, Jan 12 2019
%o A192911 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (x+1)*(2*x^2+2*x-1)/(x^6-x^5+2*x^4+5*x^3+4*x^2+x-1) )); // _G. C. Greubel_, Jan 12 2019
%o A192911 (Sage) ((x+1)*(2*x^2+2*x-1)/(x^6-x^5+2*x^4+5*x^3+4*x^2+x-1)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 12 2019
%o A192911 (GAP) a:=[1,0,0,3,5,16];; 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 A192911 Cf. A192232, A192744, A192912, A192913.
%K A192911 nonn,easy
%O A192911 0,4
%A A192911 _Clark Kimberling_, Jul 12 2011