A192878 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

3, 0, 4, 5, 18, 42, 115, 296, 780, 2037, 5338, 13970, 36579, 95760, 250708, 656357, 1718370, 4498746, 11777875, 30834872, 80726748, 211345365, 553309354, 1448582690, 3792438723, 9928733472, 25993761700, 68052551621, 178163893170, 466439127882, 1221153490483

Offset: 0

Author

Clark Kimberling, Jul 11 2011

Keywords

• nonn
• easy

Comments

The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x + 1, and p(n,x) = x*p(n-1,x) + 2*(x^2)*p(n-1,x) + 1. See A192872.

For n>0, also the coefficient of x in the reduction x^2 -> x + 1 of the polynomial A000285(n-1)*x^(n-1). - R. J. Mathar, Jul 12 2011

Examples

			The first six polynomials and reductions:
p(0,x) = 3 -> 3
p(1,x) = x -> x
p(2,x) = 4*x^2 -> 4 + 4*x
p(3,x) = 5*x^3 -> 5 + 10*x
p(4,x) = 9*x^4 -> 18 + 27*x
p(5,x) = 14*x^5 -> 42 + 27*x
In general, p(n,x) = A104449(n)*x^n -> A192878(n) + A192879(n)*x.
		

Links

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

Crossrefs

Cf. A000285, A192872, A192879.

Programs

GAP

a:=[3,0,4];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jan 08 2019

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (3-2*x^2-6*x)/((1+x)*(x^2-3*x+1)) )); // G. C. Greubel, Jan 08 2019

Mathematica

q = x^2; s = x + 1; z = 25;
p[0, x_] := 3; 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}]  (* A192878 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]  (* A192879 *)
FindLinearRecurrence[u1]
FindLinearRecurrence[u2]
LinearRecurrence[{2,2,-1}, {3,0,4}, 30] (* G. C. Greubel, Jan 08 2019 *)

PARI

a(n) = round((2^(-n)*(7*(-2)^n-(-4+sqrt(5))*(3+sqrt(5))^n+(3-sqrt(5))^n*(4+sqrt(5))))/5) \\ Colin Barker, Sep 29 2016

PARI

Vec((3-2*x^2-6*x)/((1+x)*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Sep 29 2016

Sage

((3-2*x^2-6*x)/((1+x)*(x^2-3*x+1))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 08 2019

Formula

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

G.f.: ( 3 - 2*x^2 - 6*x ) / ( (1+x)*(1 - 3*x + x^2) ). - R. J. Mathar, May 07 2014

a(n) = (2^(-n)*(7*(-2)^n - (-4+sqrt(5))*(3+sqrt(5))^n + (3-sqrt(5))^n*(4+sqrt(5))))/5. - Colin Barker, Sep 29 2016