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

Author

Clark Kimberling, Jul 14 2011

Keywords

• nonn
• easy

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

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

Crossrefs

Cf. A192232, A192744, A192951, A193009.

Programs

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 *)

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]