A192805 Constant term in the reduction of the polynomial 1+x+x^2+...+x^n by x^3->x^2+2x+1. See Comments.

1, 1, 1, 2, 3, 6, 12, 25, 53, 113, 242, 519, 1114, 2392, 5137, 11033, 23697, 50898, 109323, 234814, 504356, 1083305, 2326829, 4997793, 10734754, 23057167, 49524466, 106373552, 228479649, 490751217, 1054084065, 2264066146, 4862985491

Offset

0,4

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Links

Table of n, a(n) for n=0..32.

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

Formula

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

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

a(n)-a(n-1) = A002478(n-3). - R. J. Mathar, May 06 2014

Example

The first five polynomials p(n,x) and their reductions:
p(1,x)=1 -> 1
p(2,x)=x+1 -> x+1
p(3,x)=x^2+x+1 -> x^2+x+1
p(4,x)=x^3+x^2+x+1 -> 2x^2+3x+2
p(5,x)=x^4+x^3+x^2+x+1 -> 5x^2+6*x+3, so that
A192805=(1,1,1,2,3,...), A002478=(0,1,1,3,6,...), A077864=(0,0,1,2,5,...).

Mathematica

q = x^3; s = x^2 + 2 x + 1; z = 40;
p[0, x_] := 1; p[n_, x_] := x^n + p[n - 1, x];
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}]
  (* A192805 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A002478  *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
  (* A077864 *)

Crossrefs

Cf. A192744, A192232, A002478.

Sequence in context: A116380 A004111 A032235 * A162985 A052523 A262430

Adjacent sequences: A192802 A192803 A192804 * A192806 A192807 A192808

Keyword

nonn

Author

Clark Kimberling, Jul 10 2011

Status

approved