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A192798
Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2. See Comments.
6
1, 0, 1, 2, 3, 10, 17, 42, 87, 188, 411, 876, 1907, 4100, 8863, 19134, 41289, 89174, 192459, 415542, 897049, 1936576, 4180809, 9025544, 19484825, 42064320, 90809993, 196043706, 423225563, 913674090, 1972469945, 4258235410, 9192822255
OFFSET
1,4
COMMENTS
For discussions of polynomial reduction, see A192232 and A192744.
FORMULA
a(n) = a(n-1)+3*a(n-2)-3*a(n-4)+a(n-5)+a(n-6).
G.f.: -x*(x-1)*(x+1)*(x^2+x-1)/(x^6+x^5-3*x^4+3*x^2+x-1). [Colin Barker, Jul 27 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+2
F5(x)=x^4+3x^2+1 -> 4x^2+2*x+3, so that
A192798=(1,0,1,2,3,...), A192799=(0,1,0,2,2,...), A192800=(0,0,1,1,4,...)
MATHEMATICA
q = x^3; s = x^2 + 2; 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}] (* A192798 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
(* A192799 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
(* A192800 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 10 2011
EXTENSIONS
Comment in Mathematica code corrected by Colin Barker, Jul 27 2012
STATUS
approved