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 A207869 Z(n,-1), where Z(n,x) is the n-th Zeckendorf polynomial. 2

%I

%S 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,

%T 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,

%U 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

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

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

%e 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.

%t fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]],

%t t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k],

%t AppendTo[fr, 1]; t = t - Fibonacci[k],

%t AppendTo[fr, 0]]; k--]; fr]; t = Table[fb[n],

%t {n, 1, 500}];

%t b[n_] := Reverse[Table[x^k, {k, 0, n}]]

%t p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]

%t Table[p[n, x], {n, 1, 40}]

%t Table[p[n, x] /. x -> 1, {n, 1, 120}] (* A007895 *)

%t Table[p[n, x] /. x -> 2, {n, 1, 120}] (* A003714 *)

%t Table[p[n, x] /. x -> 3, {n, 1, 120}] (* A060140 *)

%t t1 = Table[p[n, x] /. x -> -1,

%t {n, 1, 420}] (* A207869 *)

%t Flatten[Position[t1, 0]] (* A207870 *)

%t t2 = Table[p[n, x] /. x -> I, {n, 1, 420}];

%t Flatten[Position[t2, 0] (* A207871 *)

%t Denominator[Table[p[n, x] /. x -> 1/2,

%t {n, 1, 120}]] (* A207872 *)

%t Numerator[Table[p[n, x] /. x -> 1/2,

%t {n, 1, 120}]] (* A207873 *)

%Y Cf. A207813, A207870.

%K sign

%O 1,4

%A _Clark Kimberling_, Feb 21 2012

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Last modified June 23 09:07 EDT 2021. Contains 345397 sequences. (Running on oeis4.)