A192748 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

0, 1, 4, 11, 24, 47, 86, 151, 258, 433, 718, 1181, 1932, 3149, 5120, 8311, 13476, 21835, 35362, 57251, 92670, 149981, 242714, 392761, 635544, 1028377, 1663996, 2692451, 4356528, 7049063, 11405678, 18454831, 29860602, 48315529, 78176230

Offset

1,3

Comments

The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+3n for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.

Links

Table of n, a(n) for n=1..35.

Formula

Conjecture: G.f.: -x^2*(1+x+x^2) / ( (x^2+x-1)*(x-1)^2 ), so the first differences are in A154691. - R. J. Mathar, May 04 2014

Mathematica

q = x^2; s = x + 1; z = 40;
p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 3 n;
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}]
  (* A154691 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192748 *)

Crossrefs

Cf. A192744, A192232.

Sequence in context: A057304 A001752 A160860 * A143075 A290707 A322618

Adjacent sequences: A192745 A192746 A192747 * A192749 A192750 A192751

Keyword

nonn

Author

Clark Kimberling, Jul 09 2011

Status

approved