A192812 Constant term in the reduction of the polynomial x^(2*n) + x^n + 1 by x^3 -> x + 1. See Comments.

3, 1, 1, 3, 3, 5, 7, 11, 19, 31, 53, 91, 157, 273, 475, 829, 1449, 2535, 4439, 7777, 13631, 23899, 41911, 73511, 128953, 226231, 396921, 696433, 1222003, 2144265, 3762657, 6602651, 11586379, 20332061, 35679463, 62612011, 109874987, 192815263

Offset

0,1

Comments

For discussions of polynomial reduction, 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,-2,0,-1,1,-1,1).

Formula

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-4) + a(n-5) - a(n-6) + a(n-7).

G.f.: (3 -8*x +4*x^2 -x^4 +x^6 +2*x^3) / ((1-x)*(x^3+x^2-1)*(x^3-x^2+2*x -1)). - R. J. Mathar, May 06 2014

Maple

seq(coeff(series((3-8*x+4*x^2-x^4+x^6+2*x^3)/((1-x)*(x^3+x^2-1)*(x^3-x^2+2*x-1)), x, n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Jan 03 2019

Mathematica

q = x^3; s = x + 1; z = 40;
p[n_, x_] := x^(2 n) + x^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}] (* A192812 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192813 *)
LinearRecurrence[{3, -2, 0, -1, 1, -1, 1}, {3, 1, 1, 3, 3, 5, 7}, 30] (* G. C. Greubel, Jan 03 2019 *)

Prog

(PARI) my(x='x+O('x^40)); Vec((3-8*x+4*x^2-x^4+x^6+2*x^3)/((1-x)*(1-x^2-x^3)*(1-2*x+x^2-x^3))) \\ G. C. Greubel, Jan 03 2019
(Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (3-8*x+4*x^2-x^4+x^6+2*x^3)/((1-x)*(1-x^2-x^3)*(1-2*x+x^2-x^3)) )); // G. C. Greubel, Jan 03 2019
(Sage) ((3-8*x+4*x^2-x^4+x^6+2*x^3)/((1-x)*(1-x^2-x^3)*(1-2*x+x^2-x^3)) ).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jan 03 2019

Crossrefs

Cf. A192744, A192232, A192813.

Sequence in context: A230293 A133116 A059959 * A125127 A051120 A114476

Adjacent sequences: A192809 A192810 A192811 * A192813 A192814 A192815

Keyword

nonn

Author

Clark Kimberling, Jul 10 2011

Status

approved