

A192958


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


3



1, 1, 3, 7, 17, 33, 61, 107, 183, 307, 509, 837, 1369, 2231, 3627, 5887, 9545, 15465, 25045, 40547, 65631, 106219, 171893, 278157, 450097, 728303, 1178451, 1906807, 3085313, 4992177, 8077549, 13069787, 21147399, 34217251, 55364717, 89582037
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OFFSET

0,3


COMMENTS

The titular polynomials are defined recursively: p(n,x)=x*p(n1,x)2+n^2, 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..35.
Index to sequences with linear recurrences with constant coefficients, signature (3,2,1,1).


FORMULA

a(n) = 3*a(n1)  2*a(n2)  a(n3) + a(n4).
G.f.: ( 1+4*x8*x^2+3*x^3 ) / ( (x^2+x1)*(x1)^2 ).  R. J. Mathar, May 09 2014
a(n)  2*a(n1) +a(n2) = A022089(n3).  R. J. Mathar, May 09 2014


MATHEMATICA

q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n  1, x] + n^2  2;
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}]
(* A192958 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
(* A192959 *)


CROSSREFS

Cf. A192232, A192744, A192951, A192959.
Sequence in context: A233930 A176690 A168582 * A219293 A178521 A034482
Adjacent sequences: A192955 A192956 A192957 * A192959 A192960 A192961


KEYWORD

sign


AUTHOR

Clark Kimberling, Jul 13 2011


STATUS

approved



