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A192880 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

%I A192880 #23 Dec 12 2023 08:28:44
%S A192880 1,0,3,7,34,123,495,1912,7501,29253,114342,446545,1744489,6814224,
%T A192880 26618619,103979239,406172770,1586623227,6197795703,24210320296,
%U A192880 94572284197,369425778645,1443080391558,5637075481729,22019992977457
%N A192880 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.
%C A192880 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 A192880 G. C. Greubel, <a href="/A192880/b192880.txt">Table of n, a(n) for n = 0..1000</a>
%H A192880 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,7,2,-1).
%F A192880 a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4).
%F A192880 G.f.: (1+x)*(1-3*x-x^2) / ( 1-2*x-7*x^2-2*x^3+x^4 ). - _R. J. Mathar_, May 07 2014
%F A192880 a(n) = Fibonacci(n-1)*Pell-Lucas(n)/2, where Pell-Lucas(n) = A002203(n). - _G. C. Greubel_, Jul 29 2019
%t A192880 (* First program *)
%t A192880 q = x^2; s = x + 1; z = 25;
%t A192880 p[0, x_]:= 1; p[1, x_]:= x;
%t A192880 p[n_, x_]:= 2 p[n-1, x]*x + p[n-2, x]*x^2;
%t A192880 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192880 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192880 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192880 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192880 *)
%t A192880 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192882 *)
%t A192880 FindLinearRecurrence[u1]
%t A192880 FindLinearRecurrence[u2]
%t A192880 (* Additional programs *)
%t A192880 LinearRecurrence[{2,7,2,-1}, {1,0,3,7}, 30] (* _G. C. Greubel_, Jan 08 2019 *)
%t A192880 Table[Fibonacci[n-1]*LucasL[n, 2]/2, {n,0,30}] (* _G. C. Greubel_, Jul 29 2019 *)
%o A192880 (PARI) my(x='x+O('x^30)); Vec((1+x)*(1-3*x-x^2)/(1-2*x-7*x^2-2*x^3+x^4)) \\ _G. C. Greubel_, Jan 08 2019
%o A192880 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)*(1-3*x-x^2)/(1-2*x-7*x^2-2*x^3+x^4) )); // _G. C. Greubel_, Jan 08 2019
%o A192880 (SageMath) ((1+x)*(1-3*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 A192880 (SageMath) [fibonacci(n-1)*lucas_number2(n, 2, -1)/2 for n in (0..30)] # _G. C. Greubel_, Jul 29 2019
%o A192880 (GAP) a:=[1,0,3,7];; 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 A192880 Cf. A000045, A002203, A192872, A192882.
%K A192880 nonn
%O A192880 0,3
%A A192880 _Clark Kimberling_, Jul 11 2011