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

Author

Clark Kimberling, Jul 10 2011

Keywords

• nonn

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Examples

			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,...).
		

Links

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

Crossrefs

Cf. A192744, A192232, A002478.

Programs

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 *)

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