A192780 -id:A192780

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A192781

Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+1.

+10
3

0, 1, 0, 2, 1, 4, 6, 12, 25, 46, 96, 183, 368, 720, 1424, 2809, 5536, 10930, 21545, 42516, 83846, 165404, 326257, 643550, 1269440, 2503983, 4939232, 9742752, 19217952, 37908017, 74774848, 147495906, 290940561, 573890084, 1132017286, 2232942124

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OFFSET

1,4

COMMENTS

For discussions of polynomial reduction, see A192232 and A192744.

LINKS

Table of n, a(n) for n=1..36.

Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-3,1,1).

FORMULA

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

G.f.: x^2*(x^2+x-1)/(x^6+x^5-3*x^4-x^3+3*x^2+x-1). [Colin Barker, Nov 23 2012]

EXAMPLE

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

F1(x)=1 -> 1

F2(x)=x -> x

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

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

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

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

MATHEMATICA

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

(* A192780 *)

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

(* A192781 *)

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

(* A192782 *)

CROSSREFS

Cf. A192744, A192232, A192616, A192780, A192782.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 09 2011

STATUS

approved

A192782