A192798 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2. See Comments.

1, 0, 1, 2, 3, 10, 17, 42, 87, 188, 411, 876, 1907, 4100, 8863, 19134, 41289, 89174, 192459, 415542, 897049, 1936576, 4180809, 9025544, 19484825, 42064320, 90809993, 196043706, 423225563, 913674090, 1972469945, 4258235410, 9192822255

Offset

1,4

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Links

Table of n, a(n) for n=1..33.

Index entries for linear recurrences with constant coefficients, signature (1,3,0,-3,1,1).

Formula

a(n) = a(n-1)+3*a(n-2)-3*a(n-4)+a(n-5)+a(n-6).

G.f.: -x*(x-1)*(x+1)*(x^2+x-1)/(x^6+x^5-3*x^4+3*x^2+x-1). [Colin Barker, Jul 27 2012]

Example

The first five polynomials p(n,x) and their reductions:
F1(x)=1 -> 1
F2(x)=x -> x
F3(x)=x^2+1 -> x^2+1
F4(x)=x^3+2x -> x^2+2x+2
F5(x)=x^4+3x^2+1 -> 4x^2+2*x+3, so that
A192798=(1,0,1,2,3,...), A192799=(0,1,0,2,2,...), A192800=(0,0,1,1,4,...)

Mathematica

q = x^3; s = x^2 + 2; z = 40;
p[n_, x_] := Fibonacci[n, x];
Table[Expand[p[n, x]], {n, 1, 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, 1, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]  (* A192798 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192799 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
  (* A192800 *)

Crossrefs

Cf. A192744, A192232, A192616, A192799, A192800.

Sequence in context: A328343 A309350 A300285 * A143609 A363430 A350913

Adjacent sequences: A192795 A192796 A192797 * A192799 A192800 A192801

Keyword

nonn,easy

Author

Clark Kimberling, Jul 10 2011

Extensions

Comment in Mathematica code corrected by Colin Barker, Jul 27 2012

Status

approved