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A192777 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+3x+1. See Comments. 6
1, 0, 1, 1, 2, 8, 14, 55, 121, 392, 989, 2912, 7797, 22104, 60553, 169289, 467622, 1300888, 3603914, 10008543, 27755249, 77034176, 213702153, 593005504, 1645265209, 4565154816, 12666317073, 35144684065, 97512548090, 270561677224 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

For discussions of polynomial reduction, see A192232 and A192744.

LINKS

Table of n, a(n) for n=1..30.

Index entries for linear recurrences with constant coefficients, signature (1,6,-1,-6,1,1).

FORMULA

a(n)=a(n-1)+6*a(n-2)-a(n-3)-6*a(n-4)+a(n-5)+a(n-6).

G.f.: -x*(1-5*x^2+x^4-x+x^3) / ( (x^2-x-1)*(x^4+2*x^3-3*x^2-2*x+1) ). - R. J. Mathar, May 06 2014

EXAMPLE

The first five polynomials p(n,x) and their reductions are as follows:

F1(x)=1 -> 1

F2(x)=x -> x

F3(x)=x^2+1 -> x^2+1

F4(x)=x^3+2x -> x^2+5x+1

F5(x)=x^4+3x^2+1 -> 7x^2+4x+2, so that

A192777=(1,0,1,1,2,...), A192778=(0,1,0,5,4,...), A192779=(0,0,1,1,7,...)

MATHEMATICA

q = x^3; s = x^2 + 3 x + 1; z = 40;

p[n_, x_] := Fibonacci[n, x];

Table[Expand[p[n, x]], {n, 1, 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, 1, z}];

u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]

  (* A192777 *)

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

  (* A192778 *)

u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]

  (* A192779 *)

CROSSREFS

Cf. A192744, A192232, A192616, A192772, A192778, A192779.

Sequence in context: A111001 A055258 A277649 * A054981 A059449 A272930

Adjacent sequences:  A192774 A192775 A192776 * A192778 A192779 A192780

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 09 2011

STATUS

approved

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Last modified October 22 02:35 EDT 2020. Contains 337948 sequences. (Running on oeis4.)