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A192973 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

%I A192973 #26 Sep 08 2022 08:45:58
%S A192973 1,3,10,23,47,88,157,271,458,763,1259,2064,3369,5483,8906,14447,23415,
%T A192973 37928,61413,99415,160906,260403,421395,681888,1103377,1785363,
%U A192973 2888842,4674311,7563263,12237688,19801069,32038879,51840074,83879083
%N A192973 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C A192973 The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 1 +2*n^2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.
%H A192973 Vincenzo Librandi, <a href="/A192973/b192973.txt">Table of n, a(n) for n = 1..1000</a>
%H A192973 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A192973 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
%F A192973 G.f.: x*(1+3*x^2)/((1-x-x^2)*(1-x)^2). - _R. J. Mathar_, May 11 2014
%F A192973 a(n) = Lucas(n+4) - Fibonacci(n-1) - 2*(2*n+3). - _Ehren Metcalfe_, Jul 13 2019
%t A192973 (* First program *)
%t A192973 q = x^2; s = x + 1; z = 40;
%t A192973 p[0, x]:= 1;
%t A192973 p[n_, x_]:= x*p[n-1, x] + 2*n^2 +1;
%t A192973 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192973 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192973 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192973 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192973 *)
%t A192973 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192974 *)
%t A192973 (* Additional programs *)
%t A192973 LinearRecurrence[{3, -2, -1, 1}, {1, 3, 10, 23}, 50] (* _Vincenzo Librandi_, Jul 14 2019 *)
%t A192973 With[{F = Fibonacci}, Table[F[n+4]+3*F[n+2] -2*(2*n+3), {n,40}]] (* _G. C. Greubel_, Jul 24 2019 *)
%o A192973 (Magma) [Lucas(n+4)-Fibonacci(n-1)-2*(2*n+3): n in [1..40]]; // _Vincenzo Librandi_, Jul 14 2019
%o A192973 (PARI) vector(40, n, f=fibonacci; f(n+4)+3*f(n+2) -2*(2*n+3)) \\ _G. C. Greubel_, Jul 24 2019
%o A192973 (Sage) f=fibonacci; [f(n+4)+3*f(n+2) -2*(2*n+3) for n in (1..40)] # _G. C. Greubel_, Jul 24 2019
%o A192973 (GAP) F:=Fibonacci;; List([1..40], n-> F(n+4)+3*F(n+2) -2*(2*n+3)); # _G. C. Greubel_, Jul 24 2019
%Y A192973 Cf. A000032, A000045, A192232, A192744, A192951, A192974.
%K A192973 nonn,easy
%O A192973 1,2
%A A192973 _Clark Kimberling_, Jul 13 2011