A192758 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

0, 1, 2, 4, 7, 13, 22, 37, 61, 101, 165, 269, 437, 710, 1151, 1865, 3020, 4890, 7915, 12810, 20730, 33546, 54282, 87834, 142122, 229963, 372092, 602062, 974161, 1576231, 2550400, 4126639, 6677047, 10803695, 17480751, 28284455, 45765215

Offset: 1

Author

Clark Kimberling, Jul 09 2011

Keywords

• nonn

Comments

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

Crossrefs

Cf. A192744, A192232.

Programs

Mathematica

q = x^2; s = x + 1; z = 40;
p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + Floor[(n + 4)/4] /; n > 0;
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}]
  (* A080239 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192758 *)

Formula

Conjecture: G.f.: -x^2 / ( (1+x)*(x^2+1)*(x^2+x-1)*(x-1)^2 ), partial sums of A080239. a(n)-a(n-2) = A097083(n-1). - R. J. Mathar, May 04 2014