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A192917 Coefficient of x in the reduction by (x^2 -> x+1) of the polynomial C(n)*x^n, where C=A022095.

%I A192917 #16 Sep 08 2022 08:45:58
%S A192917 0,5,6,22,51,140,360,949,2478,6494,16995,44500,116496,304997,798486,
%T A192917 2090470,5472915,14328284,37511928,98207509,257110590,673124270,
%U A192917 1762262211,4613662372,12078724896,31622512325,82788812070,216743923894,567442959603,1485584954924
%N A192917 Coefficient of x in the reduction by (x^2 -> x+1) of the polynomial C(n)*x^n, where C=A022095.
%C A192917 See A192872.
%H A192917 Colin Barker, <a href="/A192917/b192917.txt">Table of n, a(n) for n = 0..1000</a>
%H A192917 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A192917 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
%F A192917 G.f.: x*(5 -4*x)/((1+x)*(1-3*x+x^2)). - _R. J. Mathar_, May 08 2014
%F A192917 a(n) = -3*a(n-1) +a(n-2) = 9*(-1)^(n+1). - _R. J. Mathar_, May 08 2014
%F A192917 a(n) = (2^(-1-n)*(-9*(-1)^n*2^(1+n)-(3-sqrt(5))^n*(-9+sqrt(5))+(3+sqrt(5))^n*(9+sqrt(5))))/5. - _Colin Barker_, Oct 01 2016
%F A192917 a(n) = Fibonacci(2*n+1) + 2*Fibonacci(n)^2 - (-1)^n. - _G. C. Greubel_, Jul 29 2019
%t A192917 (* First program *)
%t A192917 q = x^2; s = x + 1; z = 28;
%t A192917 p[0, x_]:= 1; p[1, x_]:= 5 x;
%t A192917 p[n_, x_]:= p[n-1, x]*x + p[n-2, x]*x^2;
%t A192917 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192917 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192917 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192917 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192914 *)
%t A192917 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* see A192878 *)
%t A192917 (* Second program *)
%t A192917 With[{F=Fibonacci}, Table[F[2*n+1] +2*F[n]^2 -(-1)^n, {n,0,30}]] (* _G. C. Greubel_, Jul 28 2019 *)
%o A192917 (PARI) a(n) = round((2^(-1-n)*(-9*(-1)^n*2^(1+n)-(3-sqrt(5))^n*(-9+sqrt(5))+(3+sqrt(5))^n*(9+sqrt(5))))/5) \\ _Colin Barker_, Oct 01 2016
%o A192917 (PARI) concat(0, Vec((-x*(-5+4*x))/((1+x)*(x^2-3*x+1)) + O(x^40))) \\ _Colin Barker_, Oct 01 2016
%o A192917 (PARI) vector(30, n, n--; f=fibonacci; f(2*n+1) +2*f(n)^2 -(-1)^n) \\ _G. C. Greubel_, Jul 29 2019
%o A192917 (Magma) F:=Fibonacci; [F(2*n+1) +2*F(n)^2 -(-1)^n: n in [0..30]]; // _G. C. Greubel_, Jul 29 2019
%o A192917 (Sage) f=fibonacci; [f(2*n+1) +2*f(n)^2 -(-1)^n for n in (0..30)] # _G. C. Greubel_, Jul 29 2019
%o A192917 (GAP) F:=Fibonacci;; List([0..30], n-> F(2*n+1) +2*F(n)^2 -(-1)^n); # _G. C. Greubel_, Jul 29 2019
%Y A192917 Cf. A000045, A192232, A192744, A192872, A192916.
%K A192917 nonn,easy
%O A192917 0,2
%A A192917 _Clark Kimberling_, Jul 12 2011