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A192914 Constant term in the reduction by (x^2 -> x + 1) of the polynomial C(n)*x^n, where C=A000285.

%I A192914 #29 Sep 08 2022 08:45:58
%S A192914 1,0,5,9,28,69,185,480,1261,3297,8636,22605,59185,154944,405653,
%T A192914 1062009,2780380,7279125,19057001,49891872,130618621,341963985,
%U A192914 895273340,2343856029,6136294753,16065028224,42058789925,110111341545,288275234716,754714362597
%N A192914 Constant term in the reduction by (x^2 -> x + 1) of the polynomial C(n)*x^n, where C=A000285.
%C A192914 See A192872.
%H A192914 Colin Barker, <a href="/A192914/b192914.txt">Table of n, a(n) for n = 0..1000</a>
%H A192914 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A192914 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
%F A192914 G.f.: (1 + 3*x^2 - 2*x)/((1 + x)*(x^2 - 3*x + 1)). - _R. J. Mathar_, May 08 2014
%F A192914 a(n) = (2^(-1-n)*(3*(-1)^n*2^(2+n) + (3 + sqrt(5))^n*(-1 + 3*sqrt(5)) - (3-sqrt(5))^n*(1 + 3*sqrt(5))))/5. - _Colin Barker_, Sep 29 2016
%F A192914 a(n) = F(n+1)^2 + F(n)*F(n-3). - _Bruno Berselli_, Feb 15 2017
%t A192914 q = x^2; s = x + 1; z = 28;
%t A192914 p[0, x_]:= 1; p[1, x_]:= 4 x;
%t A192914 p[n_, x_] := p[n-1, x]*x + p[n-2, x]*x^2;
%t A192914 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192914 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192914 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192914 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192914 *)
%t A192914 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* see A192878 *)
%t A192914 LinearRecurrence[{2,2,-1}, {1,0,5}, 30] (* or *) With[{F:= Fibonacci}, Table[F[n+1]^2 +F[n]*F[n-3], {n, 0, 30}]] (* _G. C. Greubel_, Jan 12 2019 *)
%o A192914 (PARI) a(n) = round((2^(-1-n)*(3*(-1)^n*2^(2+n)+(3+sqrt(5))^n*(-1+3*sqrt(5))-(3-sqrt(5))^n*(1+3*sqrt(5))))/5) \\ _Colin Barker_, Sep 29 2016
%o A192914 (PARI) Vec((1+3*x^2-2*x)/((1+x)*(x^2-3*x+1)) + O(x^30)) \\ _Colin Barker_, Sep 29 2016
%o A192914 (PARI) {f=fibonacci}; vector(30, n, n--; f(n+1)^2 +f(n)*f(n-3)) \\ _G. C. Greubel_, Jan 12 2019
%o A192914 (Magma) F:=Fibonacci; [F(n+1)^2+F(n)*F(n-3): n in [0..30]]; // _Bruno Berselli_, Feb 15 2017
%o A192914 (Sage) f=fibonacci; [f(n+1)^2 +f(n)*f(n-3) for n in (0..30)] # _G. C. Greubel_, Jan 12 2019
%o A192914 (GAP) F:=Fibonacci; List([0..30], n -> F(n+1)^2 +F(n)*F(n-3)); # _G. C. Greubel_, Jan 12 2019
%Y A192914 Cf. A192232, A192744, A192872.
%K A192914 nonn,easy
%O A192914 0,3
%A A192914 _Clark Kimberling_, Jul 12 2011