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A192962 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

%I A192962 #23 Jun 27 2025 22:38:26
%S A192962 1,2,7,15,30,55,97,166,279,463,762,1247,2033,3306,5367,8703,14102,
%T A192962 22839,36977,59854,96871,156767,253682,410495,664225,1074770,1739047,
%U A192962 2813871,4552974,7366903,11919937,19286902,31206903,50493871,81700842
%N A192962 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C A192962 The titular polynomials are defined recursively:  p(n,x) = x*p(n-1,x) + n + 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 A192962 Vincenzo Librandi, <a href="/A192962/b192962.txt">Table of n, a(n) for n = 1..1000</a>
%H A192962 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A192962 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
%F A192962 From _R. J. Mathar_, May 09 2014: (Start)
%F A192962 G.f.: x*(1 -x +3*x^2 -x^3)/((1-x-x^2)*(1-x)^2).
%F A192962 a(n) -2*a(n-1) + a(n-2) = A022120(n-4). (End)
%F A192962 a(n) = 3*Fibonacci(n+1) + 4*Fibonacci(n) - 2*(n+2). - _G. C. Greubel_, Jul 12 2019
%t A192962 (* First program *)
%t A192962 q = x^2; s = x + 1; z = 40;
%t A192962 p[0, x]:= 1;
%t A192962 p[n_, x_]:= x*p[n-1, x] + n(n+1);
%t A192962 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192962 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192962 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192962 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192962 *)
%t A192962 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192963 *)
%t A192962 (* Additional programs *)
%t A192962 CoefficientList[Series[(1-x+3x^2-x^3)/((1-x-x^2)(1-x)^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, May 09 2014 *)
%t A192962 With[{F=Fibonacci}, Table[3*F[n+1]+4*F[n] -2*(n+2), {n,1,40}]] (* _G. C. Greubel_, Jul 12 2019 *)
%o A192962 (PARI) vector(40, n, f=fibonacci; 3*f(n+1)+4*f(n)-2*(n+2)) \\ _G. C. Greubel_, Jul 12 2019
%o A192962 (Magma) F:=Fibonacci; [3*F(n+1) +4*F(n) -2*(n+2): n in [1..40]]; // _G. C. Greubel_, Jul 12 2019
%o A192962 (Sage) f=fibonacci; [3*f(n+1) +4*f(n) -2*(n+2) for n in (1..40)] # _G. C. Greubel_, Jul 12 2019
%o A192962 (GAP) F:=Fibonacci;; List([1..40], n-> 3*F(n+1) +4*F(n) -2*(n+2)); # _G. C. Greubel_, Jul 12 2019
%Y A192962 Cf. A000045, A192232, A192744, A192951, A192963.
%K A192962 nonn
%O A192962 1,2
%A A192962 _Clark Kimberling_, Jul 13 2011