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A192952 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

%I A192952 #13 Sep 08 2022 08:45:58
%S A192952 0,1,2,7,16,33,62,111,192,325,542,895,1468,2397,3902,6339,10284,16669,
%T A192952 27002,43723,70780,114561,185402,300027,485496,785593,1271162,2056831,
%U A192952 3328072,5384985,8713142,14098215,22811448,36909757,59721302,96631159
%N A192952 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C A192952 The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 3n - 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 A192952 G. C. Greubel, <a href="/A192952/b192952.txt">Table of n, a(n) for n = 0..1000</a>
%H A192952 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A192952 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
%F A192952 From _R. J. Mathar_, May 08 2014: (Start)
%F A192952 G.f.: x*(1 -x +3*x^2)/((1-x-x^2)*(1-x)^2).
%F A192952 a(n) - a(n-1) = A192746(n-2). (End)
%F A192952 a(n) = 4*Fibonacci(n+2) - (3*n+4). - _G. C. Greubel_, Jul 12 2019
%t A192952 (* First program *)
%t A192952 q = x^2; s = x + 1; z = 40;
%t A192952 p[0, x]:= 1;
%t A192952 p[n_, x_]:= x*p[n-1, x] + 3n - 2;
%t A192952 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192952 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192952 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192952 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192746 *)
%t A192952 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192952 *)
%t A192952 (* Second program *)
%t A192952 With[{F=Fibonacci}, Table[4*F[n+2]-(3*n+4), {n,0,40}]] (* _G. C. Greubel_, Jul 12 2019 *)
%o A192952 (PARI) vector(40, n, n--; f=fibonacci; 4*f(n+2)-(3*n+4)) \\ _G. C. Greubel_, Jul 12 2019
%o A192952 (Magma) F:=Fibonacci; [4*F(n+2)-(3*n+4): n in [0..40]]; // _G. C. Greubel_, Jul 12 2019
%o A192952 (Sage) f=fibonacci; [4*f(n+2)-(3*n+4) for n in (0..40)] # _G. C. Greubel_, Jul 12 2019
%o A192952 (GAP) F:=Fibonacci;; List([0..40], n-> 4*F(n+2)-(3*n+4)); # _G. C. Greubel_, Jul 12 2019
%Y A192952 Cf. A000045, A192232, A192744, A192951.
%K A192952 nonn
%O A192952 0,3
%A A192952 _Clark Kimberling_, Jul 13 2011