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A192964 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments. 3
1, 0, 3, 7, 16, 31, 57, 100, 171, 287, 476, 783, 1281, 2088, 3395, 5511, 8936, 14479, 23449, 37964, 61451, 99455, 160948, 260447, 421441, 681936, 1103427, 1785415, 2888896, 4674367, 7563321, 12237748, 19801131, 32038943, 51840140, 83879151 (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) - n + 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 entries for 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.: ( -1-5*x^2+x^3+3*x ) / ( (x^2+x-1)*(x-1)^2 ). - R. J. Mathar, 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;

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}]

  (* A192964 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]

  (* A192965 *)

CROSSREFS

Cf. A192232, A192744, A192951, A192965.

Sequence in context: A224340 A240739 A000412 * A179904 A161810 A084631

Adjacent sequences:  A192961 A192962 A192963 * A192965 A192966 A192967

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 13 2011

STATUS

approved

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Last modified December 3 21:14 EST 2016. Contains 278745 sequences.