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A192916 Constant term in the reduction by (x^2 -> x+1) of the polynomial C(n)*x^n, where C=A022095.

%I A192916 #21 Sep 08 2022 08:45:58
%S A192916 1,0,6,11,34,84,225,584,1534,4011,10506,27500,72001,188496,493494,
%T A192916 1291979,3382450,8855364,23183649,60695576,158903086,416013675,
%U A192916 1089137946,2851400156,7465062529,19543787424,51166299750,133955111819,350699035714,918141995316
%N A192916 Constant term in the reduction by (x^2 -> x+1) of the polynomial C(n)*x^n, where C=A022095.
%C A192916 See A192872.
%H A192916 Colin Barker, <a href="/A192916/b192916.txt">Table of n, a(n) for n = 0..1000</a>
%H A192916 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A192916 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
%F A192916 G.f.: (1 -2*x +4*x^2)/((1+x)*(1-3*x+x^2)). - _R. J. Mathar_, May 08 2014
%F A192916 a(n) + a(n+1) = A054492(n). - _R. J. Mathar_, May 07 2014
%F A192916 a(n) = (2^(-n)*(7*(-2)^n+(3+sqrt(5))^n*(-1+2*sqrt(5))-(3-sqrt(5))^n*(1+2*sqrt(5))))/5. - _Colin Barker_, Oct 01 2016
%F A192916 a(n) = Fibonacci(2*n) + 2*Fibonacci(n)*Fibonacci(n-1) + (-1)^n. - _G. C. Greubel_, Jul 28 2019
%t A192916 (* First program *)
%t A192916 q = x^2; s = x + 1; z = 28;
%t A192916 p[0, x_]:= 1; p[1, x_]:= 5 x;
%t A192916 p[n_, x_]:= p[n-1, x]*x + p[n-2, x]*x^2;
%t A192916 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192916 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192916 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192916 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192914 *)
%t A192916 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* see A192878 *)
%t A192916 (* Second program *)
%t A192916 With[{F=Fibonacci}, Table[F[2*n] +2*F[n]*F[n-1] +(-1)^n, {n,0,30}]] (* _G. C. Greubel_, Jul 28 2019 *)
%o A192916 (PARI) a(n) = round((2^(-n)*(7*(-2)^n+(3+sqrt(5))^n*(-1+2*sqrt(5))-(3-sqrt(5))^n*(1+2*sqrt(5))))/5) \\ _Colin Barker_, Oct 01 2016
%o A192916 (PARI) Vec((1+4*x^2-2*x)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ _Colin Barker_, Oct 01 2016
%o A192916 (PARI) vector(30, n, n--; f=fibonacci; f(2*n) +2*f(n)*f(n-1) +(-1)^n) \\ _G. C. Greubel_, Jul 28 2019
%o A192916 (Magma) F:=Fibonacci; [F(2*n) +2*F(n)*F(n-1) +(-1)^n: n in [0..30]]; // _G. C. Greubel_, Jul 28 2019
%o A192916 (Sage) f=fibonacci; [f(2*n) +2*f(n)*f(n-1) +(-1)^n for n in (0..30)] # _G. C. Greubel_, Jul 28 2019
%o A192916 (GAP) F:=Fibonacci;; List([0..30], n-> F(2*n) +2*F(n)*F(n-1) +(-1)^n); # _G. C. Greubel_, Jul 28 2019
%Y A192916 Cf. A000045, A192232, A192744, A192872, A192917.
%K A192916 nonn,easy
%O A192916 0,3
%A A192916 _Clark Kimberling_, Jul 12 2011