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

1,3

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.

Links

Table of n, a(n) for n=1..37.

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

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 *)

Crossrefs

Cf. A192744, A192232.

Sequence in context: A119983 A364465 A151897 * A085489 A101268 A188920

Adjacent sequences: A192755 A192756 A192757 * A192759 A192760 A192761

Keyword

nonn

Author

Clark Kimberling, Jul 09 2011

Status

approved