A192973 - OEIS

%I #26 Sep 08 2022 08:45:58

%S 1,3,10,23,47,88,157,271,458,763,1259,2064,3369,5483,8906,14447,23415,

%T 37928,61413,99415,160906,260403,421395,681888,1103377,1785363,

%U 2888842,4674311,7563263,12237688,19801069,32038879,51840074,83879083

%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) + 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 Vincenzo Librandi, <a href="/A192973/b192973.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 G.f.: x*(1+3*x^2)/((1-x-x^2)*(1-x)^2). - _R. J. Mathar_, May 11 2014

%F a(n) = Lucas(n+4) - Fibonacci(n-1) - 2*(2*n+3). - _Ehren Metcalfe_, Jul 13 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] + 2*n^2 +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}] (* A192973 *)

%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192974 *)

%t (* Additional programs *)

%t LinearRecurrence[{3, -2, -1, 1}, {1, 3, 10, 23}, 50] (* _Vincenzo Librandi_, Jul 14 2019 *)

%t With[{F = Fibonacci}, Table[F[n+4]+3*F[n+2] -2*(2*n+3), {n,40}]] (* _G. C. Greubel_, Jul 24 2019 *)

%o (Magma) [Lucas(n+4)-Fibonacci(n-1)-2*(2*n+3): n in [1..40]]; // _Vincenzo Librandi_, Jul 14 2019

%o (PARI) vector(40, n, f=fibonacci; f(n+4)+3*f(n+2) -2*(2*n+3)) \\ _G. C. Greubel_, Jul 24 2019

%o (Sage) f=fibonacci; [f(n+4)+3*f(n+2) -2*(2*n+3) for n in (1..40)] # _G. C. Greubel_, Jul 24 2019

%o (GAP) F:=Fibonacci;; List([1..40], n-> F(n+4)+3*F(n+2) -2*(2*n+3)); # _G. C. Greubel_, Jul 24 2019

%Y Cf. A000032, A000045, A192232, A192744, A192951, A192974.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Jul 13 2011