A192778 Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+3x+1.

0, 1, 0, 5, 4, 28, 48, 183, 424, 1315, 3420, 9864, 26756, 75237, 207128, 577345, 1597624, 4439764, 12307388, 34166643, 94769936, 262998791, 729644824, 2024614928, 5617339496, 15586328073, 43245649904, 119991232893, 332929027020, 923754760460, 2563067457832, 7111551574191

Offset

1,4

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Links

Paolo Xausa, Table of n, a(n) for n = 1..1000

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^2*(x^2+x-1)/((x^2-x-1)*(x^4+2*x^3-3*x^2-2*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+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

LinearRecurrence[{1, 6, -1, -6, 1, 1}, {0, 1, 0, 5, 4, 28}, 35] (* Paolo Xausa, May 19 2026 *)

Crossrefs

Cf. A192744, A192232, A192616, A192772, A192777, A192779.

Sequence in context: A275960 A024067 A361983 * A051138 A157101 A237648

Adjacent sequences: A192775 A192776 A192777 * A192779 A192780 A192781

Keyword

nonn,easy

Author

Clark Kimberling, Jul 09 2011

Extensions

More terms from Paolo Xausa, May 19 2026

Status

approved