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

Author

Clark Kimberling, Jul 12 2011

Keywords

• nonn
• easy

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.

Examples

			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
		

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).

Crossrefs

Cf. A192232, A192744, A192912, A192913.

Programs

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

Magma

m:=30; R:=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

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 *)

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

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

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