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A192772 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1. 6
1, 0, 1, 1, 2, 7, 12, 41, 86, 247, 585, 1548, 3849, 9896, 25001, 63724, 161721, 411257, 1044878, 2655719, 6748972, 17151849, 43589578, 110777391, 281529169, 715471992, 1818293377, 4620978640, 11743694657, 29845241080, 75848270001 (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..31.

Index entries for linear recurrences with constant coefficients, signature (1,5,-1,-5,1,1).

FORMULA

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

G.f.: -x*(x^2-x-1)*(x^2+2*x-1) / (x^6+x^5-5*x^4-x^3+5*x^2+x-1). [Colin Barker, Jan 17 2013]

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+4x+1

F5(x)=x^4+3x^2+1 -> 6x^2+3x+2, so that

A192772=(1,0,1,1,2,...), A192773=(0,1,0,4,3,...), A192774=(0,0,1,1,6,...)

MATHEMATICA

q = x^3; s = x^2 + 2 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}]

  (* A192772 *)

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

  (* A192773 *)

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

  (* A192774 *)

CROSSREFS

Cf. A192744, A192232, A192616, A192773, A192774.

Sequence in context: A055257 A238366 A084068 * A046243 A230302 A230637

Adjacent sequences:  A192769 A192770 A192771 * A192773 A192774 A192775

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 09 2011

STATUS

approved

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Last modified October 16 03:15 EDT 2019. Contains 328038 sequences. (Running on oeis4.)