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A192974 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

%I A192974 #28 Sep 08 2022 08:45:58
%S A192974 0,1,4,14,37,84,172,329,600,1058,1821,3080,5144,8513,13996,22902,
%T A192974 37349,60764,98692,160105,259520,420426,680829,1102224,1784112,
%U A192974 2887489,4672852,7561694,12236005,19799268,32036956,51838025,83876904,135716978
%N A192974 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C A192974 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 A192974 G. C. Greubel, <a href="/A192974/b192974.txt">Table of n, a(n) for n = 0..1000</a>
%H A192974 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).
%F A192974 a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
%F A192974 G.f.: x*(1+3*x^2)/((1-x-x^2)*(1-x)^3). - _R. J. Mathar_, May 11 2014
%F A192974 a(n) = Fibonacci(n+7) + Lucas(n+3) - 2*n*(n+4) - 17. - _Ehren Metcalfe_, Jul 14 2019
%t A192974 (* First program *)
%t A192974 q = x^2; s = x + 1; z = 40;
%t A192974 p[0, x]:= 1;
%t A192974 p[n_, x_]:= x*p[n-1, x] + 2*n^2 +1;
%t A192974 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192974 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192974 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192974 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192973 *)
%t A192974 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192974 *)
%t A192974 (* Additional programs *)
%t A192974 Table[Fibonacci[n+7] +LucasL[n+3] -2n(n+4) -17, {n,0,40}] (* _Vincenzo Librandi_, Jul 15 2019 *)
%o A192974 (PARI) a(n)=fibonacci(n+7) + fibonacci(2*n+6)/fibonacci(n+3) - 2*n*(n+4) - 17 \\ _Richard N. Smith_, Jul 14 2019
%o A192974 (Magma) [Fibonacci(n+7)+Lucas(n+3)-2*n*(n+4)-17: n in [0..40]]; // _Vincenzo Librandi_, Jul 15 2019
%o A192974 (Sage) f=fibonacci; [f(n+6)+3*f(n+4) -(2*n^2+8*n+17) for n in (0..40)] # _G. C. Greubel_, Jul 24 2019
%o A192974 (GAP) F:=Fibonacci;; List([0..40], n-> F(n+6)+3*F(n+4) -(2*n^2+8*n+17)); # _G. C. Greubel_, Jul 24 2019
%Y A192974 Cf. A000032, A000045, A192232, A192744, A192951, A192971.
%K A192974 nonn,easy
%O A192974 0,3
%A A192974 _Clark Kimberling_, Jul 13 2011