

A192979


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


3



1, 1, 4, 9, 19, 36, 65, 113, 192, 321, 531, 872, 1425, 2321, 3772, 6121, 9923, 16076, 26033, 42145, 68216, 110401, 178659, 289104, 467809, 756961, 1224820, 1981833, 3206707, 5188596, 8395361, 13584017, 21979440, 35563521, 57543027, 93106616
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OFFSET

0,3


COMMENTS

The titular polynomials are defined recursively: p(n,x)=x*p(n1,x)+1n+*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.: (3*x^22*x+1) / ((x1)^2*(x^2+x1)).  Colin Barker, May 11 2014


MATHEMATICA

q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n  1, x] + n^2  n + 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}] (* A192979 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192980 *)


CROSSREFS

Cf. A192232, A192744, A192951, A192980.
Sequence in context: A008113 A008111 A023611 * A232623 A002804 A133649
Adjacent sequences: A192976 A192977 A192978 * A192980 A192981 A192982


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Jul 13 2011


STATUS

approved



