%I #21 Sep 08 2022 08:45:58
%S 1,2,7,15,30,55,97,166,279,463,762,1247,2033,3306,5367,8703,14102,
%T 22839,36977,59854,96871,156767,253682,410495,664225,1074770,1739047,
%U 2813871,4552974,7366903,11919937,19286902,31206903,50493871,81700842
%N Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C 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 Vincenzo Librandi, <a href="/A192962/b192962.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
%F From _R. J. Mathar_, May 09 2014: (Start)
%F G.f.: x*(1 -x +3*x^2 -x^3)/((1-x-x^2)*(1-x)^2).
%F a(n) -2*a(n-1) + a(n-2) = A022120(n-4). (End)
%F a(n) = 3*Fibonacci(n+1) + 4*Fibonacci(n) - 2*(n+2). - _G. C. Greubel_, Jul 12 2019
%t (* First program *)
%t q = x^2; s = x + 1; z = 40;
%t p[0, x]:= 1;
%t p[n_, x_]:= x*p[n-1, x] + n(n+1);
%t Table[Expand[p[n, x]], {n, 0, 7}]
%t reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192962 *)
%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192963 *)
%t (* Additional programs *)
%t 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 With[{F=Fibonacci}, Table[3*F[n+1]+4*F[n] -2*(n+2), {n,1,40}]] (* _G. C. Greubel_, Jul 12 2019 *)
%o (PARI) vector(40, n, f=fibonacci; 3*f(n+1)+4*f(n)-2*(n+2)) \\ _G. C. Greubel_, Jul 1122019
%o (Magma) F:=Fibonacci; [3*F(n+1) +4*F(n) -2*(n+2): n in [1..40]]; // _G. C. Greubel_, Jul 12 2019
%o (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 (GAP) F:=Fibonacci;; List([1..40], n-> 3*F(n+1) +4*F(n) -2*(n+2)); # _G. C. Greubel_, Jul 12 2019
%Y Cf. A000045, A192232, A192744, A192951, A192963.
%K nonn
%O 1,2
%A _Clark Kimberling_, Jul 13 2011