A193008 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

1, 2, 10, 31, 78, 170, 339, 636, 1144, 1997, 3412, 5740, 9549, 15758, 25854, 42243, 68818, 111878, 181615, 294520, 477276, 773057, 1251720, 2026296, 3279673, 5307770, 8589394, 13899271, 22490934, 36392642, 58886187, 95281620, 154170784

Offset

0,2

Comments

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+1+n^3, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Links

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

Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Formula

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

G.f.: (7*x^2-2*x+1)/((x-1)^3*(x^2+x-1)). [Colin Barker, Nov 12 2012]

Mathematica

q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n - 1, x] + n^3 + 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}] (* A193008 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193009 *)

Crossrefs

Cf. A192232, A192744, A192951, A193009.

Sequence in context: A305011 A090809 A051747 * A024456 A197452 A050927

Adjacent sequences: A193005 A193006 A193007 * A193009 A193010 A193011

Keyword

nonn,easy

Author

Clark Kimberling, Jul 14 2011

Status

approved