A192762 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

0, 1, 6, 13, 26, 47, 82, 139, 232, 383, 628, 1025, 1668, 2709, 4394, 7121, 11534, 18675, 30230, 48927, 79180, 128131, 207336, 335493, 542856, 878377, 1421262, 2299669, 3720962, 6020663, 9741658, 15762355, 25504048, 41266439, 66770524

Offset: 0

Author

Clark Kimberling, Jul 09 2011

Keywords

• nonn
• easy

Comments

The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+n+4 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Crossrefs

Cf. A192744, A192232, A022319 (first differences).

Programs

Mathematica

p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 4;
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}]
  (* A022319 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192762 *)

Formula

a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). G.f.: x*(3*x^2-3*x-1) / ((x-1)^2*(x^2+x-1)). [Colin Barker, Dec 08 2012]