A192773 Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1.

0, 1, 0, 4, 3, 18, 30, 98, 219, 596, 1464, 3783, 9540, 24328, 61740, 156985, 398904, 1013772, 2576475, 6547574, 16640382, 42288806, 107473443, 273129468, 694130016, 1764047839, 4483130424, 11393354512, 28954911624, 73585574049

Offset

1,4

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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

(See A192772.)
LinearRecurrence[{1, 5, -1, -5, 1, 1}, {0, 1, 0, 4, 3, 18}, 40] (* Harvey P. Dale, Aug 07 2025 *)

Crossrefs

Cf. A192232, A192744, A192772.

Sequence in context: A373946 A276083 A161893 * A183231 A241358 A178417

Adjacent sequences: A192770 A192771 A192772 * A192774 A192775 A192776

Keyword

nonn,easy

Author

Clark Kimberling, Jul 09 2011

Status

approved