A192980 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 2, 6, 15, 34, 70, 135, 248, 440, 761, 1292, 2164, 3589, 5910, 9682, 15803, 25726, 41802, 67835, 109980, 178196, 288597, 467256, 756360, 1224169, 1981130, 3205950, 5187783, 8394490, 13583086, 21978447, 35562464, 57541904, 93105425

Offset

0,3

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 1 - 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.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Formula

a(n) = 4*a(n-1) -5*a(n-2) +a(n-3) +2*a(n-4) -a(n-5).

G.f.: x*(1-2*x+3*x^2)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 12 2014

a(n) = Fibonacci(n+4) + Lucas(n+3) - (n^2 + 3*n + 7). - G. C. Greubel, Jul 24 2019

Mathematica

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] +n^2-n+1;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192979 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192980 *)
(* Additional programs *)
CoefficientList[Series[x*(1-2*x+3*x^2)/((1-x-x^2)*(1-x)^3), {x, 0, 40}], x] (* Vincenzo Librandi, May 13 2014 *)
Table[Fibonacci[n+4]+LucasL[n+3] -(n^2+3*n+7), {n, 0, 40}] (* G. C. Greubel, Jul 24 2019 *)

Prog

(PARI) vector(40, n, n--; f=fibonacci; 2*f(n+4)+f(n+2) -(n^2+3*n+7)) \\ G. C. Greubel, Jul 24 2019
(Magma) [Fibonacci(n+4)+Lucas(n+3)-(n^2+3*n+7): n in [0..40]]; // G. C. Greubel, Jul 24 2019
(Sage) f=fibonacci; [2*f(n+4)+f(n+2) -(n^2+3*n+7) for n in (0..40)] # G. C. Greubel, Jul 24 2019
(GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+4)+F(n+2) -(n^2+3*n+7)); # G. C. Greubel, Jul 24 2019

Crossrefs

Cf. A000032, A000045, A192232, A192744, A192951, A192979.

Sequence in context: A262151 A246320 A101352 * A281970 A221719 A095380

Adjacent sequences: A192977 A192978 A192979 * A192981 A192982 A192983

Keyword

nonn

Author

Clark Kimberling, Jul 13 2011

Status

approved