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A192919 Constant term in the reduction by (x^2 -> x+1) of the polynomial F(n+4)*x^n, where F=A000045 (Fibonacci sequence).

%I A192919 #28 Sep 08 2022 08:45:58
%S A192919 3,0,8,13,42,102,275,712,1872,4893,12818,33550,87843,229968,602072,
%T A192919 1576237,4126650,10803702,28284467,74049688,193864608,507544125,
%U A192919 1328767778,3478759198,9107509827,23843770272,62423801000,163427632717,427859097162,1120149658758
%N A192919 Constant term in the reduction by (x^2 -> x+1) of the polynomial F(n+4)*x^n, where F=A000045 (Fibonacci sequence).
%C A192919 See A192872.
%H A192919 Colin Barker, <a href="/A192919/b192919.txt">Table of n, a(n) for n = 0..1000</a>
%H A192919 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A192919 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
%F A192919 a(n) = Fibonacci(n-1)*Fibonacci(n+4). - _Gary Detlefs_, Oct 19 2011
%F A192919 G.f.: (3 -6*x +2*x^2)/((1+x)*(1-3*x+x^2)). - _R. J. Mathar_, May 08 2014
%F A192919 a(n) + a(n+1) = A001906(n+1). - _R. J. Mathar_, May 08 2014
%F A192919 a(n) = (2^(-n)*(11*(-2)^n-(3-sqrt(5))^n*(-2+sqrt(5))+(2+sqrt(5))*(3+sqrt(5))^n))/5. - _Colin Barker_, Oct 01 2016
%F A192919 From _Amiram Eldar_, Oct 06 2020: (Start)
%F A192919 Sum_{n>=2} 1/a(n) = (1/5) * A290565 - 17/150.
%F A192919 Sum_{n>=2} (-1)^n/a(n) = 1/phi - 83/150, where phi is the golden ratio (A001622). (End)
%p A192919 with(combinat):seq(fibonacci(n-1)*fibonacci(n+4), n=0..27);
%t A192919 (* First program *)
%t A192919 q = x^2; s = x + 1; z = 28;
%t A192919 p[0, x_]:= 3; p[1, x_]:= 5 x;
%t A192919 p[n_, x_]:= p[n-1, x]*x + p[n-2, x]*x^2;
%t A192919 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192919 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192919 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192919 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192919 *)
%t A192919 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192920 *)
%t A192919 (* Second program *)
%t A192919 With[{F=Fibonacci}, Table[F[n-1]*F[n+4], {n,0,30}]] (* _G. C. Greubel_, Jul 28 2019 *)
%o A192919 (PARI) a(n) = round((2^(-n)*(11*(-2)^n-(3-sqrt(5))^n*(-2+sqrt(5))+(2+sqrt(5))*(3+sqrt(5))^n))/5) \\ _Colin Barker_, Oct 01 2016
%o A192919 (PARI) Vec((3+2*x^2-6*x)/((1+x)*(x^2-3*x+1)) + O(x^30)) \\ _Colin Barker_, Oct 01 2016
%o A192919 (PARI) vector(30, n, n--; f=fibonacci; f(n-1)*f(n+4)) \\ _G. C. Greubel_, Jul 28 2019
%o A192919 (Magma) F:=Fibonacci; [F(n-1)*F(n+4): n in [0..30]]; // _G. C. Greubel_, Jul 28 2019
%o A192919 (Sage) f=fibonacci; [f(n-1)*f(n+4) for n in (0..30)] # _G. C. Greubel_, Jul 28 2019
%o A192919 (GAP) F:=Fibonacci;; List([0..30], n-> F(n-1)*F(n+4)); # _G. C. Greubel_, Jul 28 2019
%Y A192919 Cf. A000045, A001622, A192232, A192744, A192872, A192883, A192920, A290565.
%K A192919 nonn,easy
%O A192919 0,1
%A A192919 _Clark Kimberling_, Jul 12 2011