A207869 Z(n,-1), where Z(n,x) is the n-th Zeckendorf polynomial.

1, -1, 1, 2, -1, 0, -2, 1, 2, 0, 2, 3, -1, 0, -2, 0, 1, -2, -1, -3, 1, 2, 0, 2, 3, 0, 1, -1, 2, 3, 1, 3, 4, -1, 0, -2, 0, 1, -2, -1, -3, 0, 1, -1, 1, 2, -2, -1, -3, -1, 0, -3, -2, -4, 1, 2, 0, 2, 3, 0, 1, -1, 2, 3, 1, 3, 4, 0, 1, -1, 1, 2, -1, 0, -2, 2, 3, 1, 3, 4, 1, 2, 0

Offset

1,4

Comments

The Zeckendorf polynomials Z(x,n) are defined and ordered at A207813.

Links

Table of n, a(n) for n=1..83.

Example

The first ten Zeckendorf polynomials are 1, x, x^2, x^2 + 1, x^3, x^3 + 1, x + x^3, x^4, 1 + x^4, x + x^4; their values at x=-1 are 1, -1, 1, 2, -1, 0, -2, 1, 2, 0, indicating initial terms for A207869 and A207870.

Mathematica

fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]],
   t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k],
    AppendTo[fr, 1]; t = t - Fibonacci[k],
    AppendTo[fr, 0]]; k--]; fr]; t = Table[fb[n],
      {n, 1, 500}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
Table[p[n, x], {n, 1, 40}]
Table[p[n, x] /. x -> 1, {n, 1, 120}]  (* A007895 *)
Table[p[n, x] /. x -> 2, {n, 1, 120}]  (* A003714 *)
Table[p[n, x] /. x -> 3, {n, 1, 120}]  (* A060140 *)
   t1 = Table[p[n, x] /. x -> -1,
   {n, 1, 420}]                        (* A207869 *)
Flatten[Position[t1, 0]]               (* A207870 *)
t2 = Table[p[n, x] /. x -> I, {n, 1, 420}];
Flatten[Position[t2, 0]                (* A207871 *)
Denominator[Table[p[n, x] /. x -> 1/2,
   {n, 1, 120}]]                       (* A207872 *)
Numerator[Table[p[n, x] /. x -> 1/2,
   {n, 1, 120}]]                       (* A207873 *)

Crossrefs

Cf. A207813, A207870.

Sequence in context: A366745 A230025 A330374 * A130210 A236459 A190427

Adjacent sequences: A207866 A207867 A207868 * A207870 A207871 A207872

Keyword

sign

Author

Clark Kimberling, Feb 21 2012

Status

approved