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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The titular polynomials are defined recursively:  p(n,x)=x*p(n-1,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(n-1) - 2*a(n-2) - a(n-3) + a(n-4).

G.f.: ( -1+4*x-8*x^2+3*x^3 ) / ( (x^2+x-1)*(x-1)^2 ). - R. J. Mathar, May 09 2014

a(n) - 2*a(n-1) +a(n-2) = A022089(n-3). - 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

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Last modified November 27 02:34 EST 2014. Contains 250152 sequences.