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

%I A207869 #16 Jun 26 2025 07:32:17
%S A207869 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 A207869 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 A207869 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 A207869 a(n) = Z(n,-1), where Z(n,x) is the n-th Zeckendorf polynomial.
%C A207869 The Zeckendorf polynomials Z(x,n) are defined and ordered at A207813.
%e A207869 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 A207869 fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]],
%t A207869    t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k],
%t A207869     AppendTo[fr, 1]; t = t - Fibonacci[k],
%t A207869     AppendTo[fr, 0]]; k--]; fr]; t = Table[fb[n],
%t A207869       {n, 1, 500}];
%t A207869 b[n_] := Reverse[Table[x^k, {k, 0, n}]]
%t A207869 p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
%t A207869 Table[p[n, x], {n, 1, 40}]
%t A207869 Table[p[n, x] /. x -> 1, {n, 1, 120}]  (* A007895 *)
%t A207869 Table[p[n, x] /. x -> 2, {n, 1, 120}]  (* A003714 *)
%t A207869 Table[p[n, x] /. x -> 3, {n, 1, 120}]  (* A060140 *)
%t A207869    t1 = Table[p[n, x] /. x -> -1,
%t A207869    {n, 1, 420}]                        (* A207869 *)
%t A207869 Flatten[Position[t1, 0]]               (* A207870 *)
%t A207869 t2 = Table[p[n, x] /. x -> I, {n, 1, 420}];
%t A207869 Flatten[Position[t2, 0]]               (* A207871 *)
%t A207869 Denominator[Table[p[n, x] /. x -> 1/2, {n, 1, 120}]]  (* A207872 *)
%t A207869 Numerator[Table[p[n, x] /. x -> 1/2, {n, 1, 120}]]    (* A207873 *)
%Y A207869 Cf. A207813, A207870.
%K A207869 sign
%O A207869 1,4
%A A207869 _Clark Kimberling_, Feb 21 2012