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A192882 Coefficient of x in the reduction by (x^2 -> x+1) of the polynomial p(n,x) given in Comments.

%I A192882 #20 Sep 08 2022 08:45:58
%S A192882 0,1,3,14,51,205,792,3107,12117,47362,184965,722591,2822544,11025793,
%T A192882 43069611,168242270,657200859,2567211037,10028243016,39173122739,
%U A192882 153021167805,597743469778,2334953116653,9120979734623,35629097057568
%N A192882 Coefficient of x in the reduction by (x^2 -> x+1) of the polynomial p(n,x) given in Comments.
%C A192882 The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x, and p(n,x) = 2*x*p(n-1,x) + (x^2)*p(n-1,x).  See A192872.
%H A192882 G. C. Greubel, <a href="/A192882/b192882.txt">Table of n, a(n) for n = 0..1000</a>
%H A192882 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,7,2,-1).
%F A192882 a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4).
%F A192882 G.f.: x*(1+x+x^2) / ( 1-2*x-7*x^2-2*x^3+x^4 ). - _R. J. Mathar_, May 07 2014
%F A192882 a(n) = Fibonacci(n)*Pell-Lucas(n)/2, where Pell-Lucas(n) = A002203(n). - _G. C. Greubel_, Jul 29 2019
%t A192882 (* First program *)
%t A192882 q = x^2; s = x + 1; z = 25;
%t A192882 p[0, x_]:= 1; p[1, x_]:= x;
%t A192882 p[n_, x_]:= 2 p[n-1, x]*x + p[n-2, x]*x^2;
%t A192882 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192882 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192882 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192882 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192880 *)
%t A192882 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192882 *)
%t A192882 FindLinearRecurrence[u1]
%t A192882 FindLinearRecurrence[u2]
%t A192882 (* Additional programs *)
%t A192882 LinearRecurrence[{2,7,2,-1}, {0,1,3,14}, 30] (* _G. C. Greubel_, Jan 08 2019 *)
%t A192882 Table[Fibonacci[n]*LucasL[n, 2]/2, {n,0,30}] (* _G. C. Greubel_, Jul 29 2019 *)
%o A192882 (PARI) my(x='x+O('x^30)); concat([0], Vec(x*(1+x+x^2)/(1-2*x-7*x^2-2*x^3 +x^4))) \\ _G. C. Greubel_, Jan 08 2019
%o A192882 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1+x+x^2)/(1-2*x-7*x^2-2*x^3+x^4) )); // _G. C. Greubel_, Jan 08 2019
%o A192882 (Sage) (x*(1+x+x^2)/(1-2*x-7*x^2-2*x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 08 2019
%o A192882 (GAP) a:=[0,1,3,14];; for n in [5..30] do a[n]:=2*a[n-1]+7*a[n-2] +2*a[n-3] -a[n-4]; od; a; # _G. C. Greubel_, Jan 08 2019
%Y A192882 Cf. A000045, A002203, A192872, A192880.
%K A192882 nonn
%O A192882 0,3
%A A192882 _Clark Kimberling_, Jul 11 2011