A207870 -id:A207870

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A207869

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

+10
2

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

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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.

KEYWORD

sign

AUTHOR

Clark Kimberling, Feb 21 2012

STATUS

approved

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