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A192616
Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+x+1. See Comments.
14
1, 0, 1, 1, 2, 6, 10, 29, 57, 142, 309, 720, 1625, 3714, 8457, 19259, 43902, 99970, 227830, 518943, 1182401, 2693624, 6136837, 13980960, 31851853, 72565704, 165320833, 376638417, 858066430, 1954869262, 4453630790, 10146374277, 23115721705
OFFSET
1,5
COMMENTS
For discussions of polynomial reduction, see A192232 and A192744.
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*(x^4+x^3-3*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:
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
q = x^3; s = x^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}]
(* A192616 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
(* A192617 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
(* A192651 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 09 2011
STATUS
approved