A192964 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

1, 0, 3, 7, 16, 31, 57, 100, 171, 287, 476, 783, 1281, 2088, 3395, 5511, 8936, 14479, 23449, 37964, 61451, 99455, 160948, 260447, 421441, 681936, 1103427, 1785415, 2888896, 4674367, 7563321, 12237748, 19801131, 32038943, 51840140, 83879151

Offset: 0

Author

Clark Kimberling, Jul 13 2011

Keywords

• nonn

Comments

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Crossrefs

Cf. A000045, A192232, A192744, A192951, A192965.

Programs

GAP

F:=Fibonacci;; List([0..40], n-> F(n+3) +3*F(n+1) -2*(n+2)); # G. C. Greubel, Jul 11 2019

Magma

F:=Fibonacci; [F(n+3) +3*F(n+1) -2*(n+2): n in [0..40]]; // G. C. Greubel, Jul 11 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(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}] (* A192964 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192965 *)
(* Second program *)
With[{F=Fibonacci}, Table[F[n+3]+3*F[n+1] -2*(n+2), {n,0,40}]] (* G. C. Greubel, Jul 11 2019 *)

PARI

vector(40, n, n--; f=fibonacci; f(n+3)+3*f(n+1)-2*(n+2)) \\ G. C. Greubel, Jul 11 2019

Sage

f=fibonacci; [f(n+3) +3*f(n+1) -2*(n+2) for n in (0..40)] # G. C. Greubel, Jul 11 2019

Formula

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

G.f.: (1 -3*x +5*x^2 -x^3)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, May 11 2014

a(n) = Fibonacci(n+3) + 3*Fibonacci(n+1) - 2*(n+2). - G. C. Greubel, Jul 11 2019