A192617 Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+x+1.

0, 1, 0, 3, 2, 10, 16, 43, 92, 213, 486, 1100, 2522, 5719, 13068, 29721, 67772, 154334, 351670, 801137, 1825184, 4158219, 9473244, 21582392, 49169220, 112018989, 255203904, 581412535, 1324587918, 3017709810, 6875021540, 15662845615

Offset

1,4

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Links

Table of n, a(n) for n=1..32.

Index entries for linear recurrences with constant coefficients, signature (1,4,-1,-4,1,1).

Formula

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

G.f.: x^2*(x^2+x-1)/(x^6+x^5-4*x^4-x^3+4*x^2+x-1). [Colin Barker, Jul 27 2012]

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+3x+1
F5(x)=x^4+3x^2+1 -> 4x^2+2x+2, so that
A192616=(1,0,1,1,2,...), A192617=(0,1,0,3,2,...), A192651=(0,0,1,1,5,...)

Mathematica

(See A192616.)
LinearRecurrence[{1, 4, -1, -4, 1, 1}, {0, 1, 0, 3, 2, 10}, 40] (* Harvey P. Dale, Feb 23 2021 *)

Crossrefs

Cf. A192744, A192232, A192616.

Sequence in context: A099946 A011953 A038519 * A082219 A034461 A070033

Adjacent sequences: A192614 A192615 A192616 * A192618 A192619 A192620

Keyword

nonn,easy

Author

Clark Kimberling, Jul 09 2011

Status

approved