A192752 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

1, 7, 12, 23, 39, 66, 109, 179, 292, 475, 771, 1250, 2025, 3279, 5308, 8591, 13903, 22498, 36405, 58907, 95316, 154227, 249547, 403778, 653329, 1057111, 1710444, 2767559, 4478007, 7245570, 11723581, 18969155, 30692740, 49661899, 80354643

Offset

0,2

Comments

The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+4n+3 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=0..34.

Formula

Conjecture: G.f.: ( 1+5*x-2*x^2 ) / ( (x-1)*(x^2+x-1) ). a(n) = A000071(n+3)+5*A000071(n+2) -2*A000071(n+1) and first differences in A022136. - 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] + 4 n + 3;
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}](* A192752 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}](* A192753 *)

Crossrefs

Cf. A192744, A192232, A192753.

Sequence in context: A273178 A160423 A252652 * A097925 A132195 A076852

Adjacent sequences: A192749 A192750 A192751 * A192753 A192754 A192755

Keyword

nonn

Author

Clark Kimberling, Jul 09 2011

Status

approved