A192772 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1.

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

Offset: 1

Author

Clark Kimberling, Jul 09 2011

Keywords

• nonn
• easy

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Examples

			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,...)
		

Links

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

Crossrefs

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

Programs

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 *)

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]