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A192958 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

%I A192958 #16 Sep 08 2022 08:45:58
%S A192958 1,-1,3,7,17,33,61,107,183,307,509,837,1369,2231,3627,5887,9545,15465,
%T A192958 25045,40547,65631,106219,171893,278157,450097,728303,1178451,1906807,
%U A192958 3085313,4992177,8077549,13069787,21147399,34217251,55364717,89582037
%N A192958 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C A192958 The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) - 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 A192958 G. C. Greubel, <a href="/A192958/b192958.txt">Table of n, a(n) for n = 0..1000</a>
%H A192958 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A192958 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
%F A192958 From _R. J. Mathar_, May 09 2014: (Start)
%F A192958 G.f.: (1 -4*x +8*x^2 -3*x^3)/((1-x-x^2)*(1-x)^2).
%F A192958 a(n) - 2*a(n-1) +a(n-2) = A022089(n-3). (End)
%F A192958 a(n) = 6*Fibonacci(n+1) - (2*n+5). - _G. C. Greubel_, Jul 12 2019
%t A192958 (* First program *)
%t A192958 q = x^2; s = x + 1; z = 40;
%t A192958 p[0, x]:= 1;
%t A192958 p[n_, x_]:= x*p[n-1, x] + n^2 - 2;
%t A192958 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192958 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192958 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192958 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192958 *)
%t A192958 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192959 *)
%t A192958 (* Second program *)
%t A192958 With[{F=Fibonacci}, Table[6*F[n+1]-(2*n+5), {n,0,40}]] (* _G. C. Greubel_, Jul 12 2019 *)
%o A192958 (PARI) vector(40, n, n--; f=fibonacci; 6*f(n+1)-(2*n+5)) \\ _G. C. Greubel_, Jul 12 2019
%o A192958 (Magma) F:=Fibonacci; [6*F(n+1)-(2*n+5): n in [0..40]]; // _G. C. Greubel_, Jul 12 2019
%o A192958 (Sage) f=fibonacci; [6*f(n+1)-(2*n+5) for n in (0..40)] # _G. C. Greubel_, Jul 12 2019
%o A192958 (GAP) F:=Fibonacci;; List([0..40], n-> 6*F(n+1)-(2*n+5)); # _G. C. Greubel_, Jul 12 2019
%Y A192958 Cf. A000045, A192232, A192744, A192951, A192959.
%K A192958 sign
%O A192958 0,3
%A A192958 _Clark Kimberling_, Jul 13 2011