A192911 Constant term in the reduction by (x^3 -> x + 1) of the polynomial F(n+1)*x^n, where F(n)=A000045 (Fibonacci sequence).

1, 0, 0, 3, 5, 16, 52, 147, 442, 1320, 3916, 11664, 34717, 103298, 307440, 914949, 2722885, 8103424, 24116008, 71769885, 213589298, 635647790, 1891705884, 5629770720, 16754357925, 49861446392, 148389084968, 441610143507

Offset

0,4

Comments

Regarding polynomial reduction, see A192232 and A192744. In the case of the reduction at A192911, each term in the three resulting sequences is a product of a Fibonacci number and a tribonacci numbers

A192911(n) = F(n+1)*T3(n+1), where F=A000045, T3=A000073.

A192912(n) = F(n+1)*T2(n), where T2=A001590.

A192913(n) = F(n+1)*T3(n).

All three obey the same linear recurrence, shown below at Formula.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Formula

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

G.f.: (x+1)*(2*x^2+2*x-1)/(x^6-x^5+2*x^4+5*x^3+4*x^2+x-1). - Colin Barker, Aug 31 2012

Example

The first six polynomials and reductions:
1 -> 1
x -> 2
2*x^2 -> 2*x^2
3*x^3 -> 3 + 3*x + 3*x^2
5*x^4 -> 5 + 10*x + 10*x^2
8*x^5 -> 16 + 24*x + 32*x^2

Mathematica

q = x^3; s = x^2 + x + 1; z = 22;
p[0, x_] := 1; p[1, x_] := x;
p[n_, x_] := p[n - 1, x]*x + p[n - 2, x]*x^2;
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}] (* A192911 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192912 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}] (* A192913 *)
LinearRecurrence[{1, 4, 5, 2, -1, 1}, {1, 0, 0, 3, 5, 16}, 28] (* Ray Chandler, Aug 02 2015 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((x+1)*(2*x^2+2*x-1)/(x^6-x^5+2*x^4+5*x^3 +4*x^2+x-1)) \\ G. C. Greubel, Jan 12 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (x+1)*(2*x^2+2*x-1)/(x^6-x^5+2*x^4+5*x^3+4*x^2+x-1) )); // G. C. Greubel, Jan 12 2019
(Sage) ((x+1)*(2*x^2+2*x-1)/(x^6-x^5+2*x^4+5*x^3+4*x^2+x-1)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 12 2019
(GAP) a:=[1, 0, 0, 3, 5, 16];; for n in [7..30] do a[n]:=a[n-1]+4*a[n-2] + 5*a[n-3]+2*a[n-4]-a[n-5]+a[n-6]; od; a; # G. C. Greubel, Jan 12 2019

Crossrefs

Cf. A192232, A192744, A192912, A192913.

Sequence in context: A099101 A208819 A038120 * A105408 A291963 A375911

Adjacent sequences: A192908 A192909 A192910 * A192912 A192913 A192914

Keyword

nonn,easy

Author

Clark Kimberling, Jul 12 2011

Status

approved