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A207870 Numbers k matched to Zeckendorf polynomials divisible by x+1.

%I A207870 #13 Jun 27 2025 00:45:18
%S A207870 6,10,14,16,23,26,35,37,42,51,57,60,68,74,83,90,92,97,106,110,116,120,
%T A207870 127,132,134,146,149,157,163,172,178,184,188,192,194,206,214,217,234,
%U A207870 236,241,250,254,260,264,271,276,278,288,294,298,302,304,311
%N A207870 Numbers k matched to Zeckendorf polynomials divisible by x+1.
%C A207870 The Zeckendorf polynomials Z(x,k) are defined and ordered at A207813.
%e A207870 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 this sequence.
%t A207870 fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]],
%t A207870    t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k],
%t A207870     AppendTo[fr, 1]; t = t - Fibonacci[k],
%t A207870     AppendTo[fr, 0]]; k--]; fr]; t = Table[fb[n],
%t A207870       {n, 1, 500}];
%t A207870 b[n_] := Reverse[Table[x^k, {k, 0, n}]]
%t A207870 p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
%t A207870 Table[p[n, x], {n, 1, 40}]
%t A207870 Table[p[n, x] /. x -> 1, {n, 1, 120}]  (* A007895 *)
%t A207870 Table[p[n, x] /. x -> 2, {n, 1, 120}]  (* A003714 *)
%t A207870 Table[p[n, x] /. x -> 3, {n, 1, 120}]  (* A060140 *)
%t A207870    t1 = Table[p[n, x] /. x -> -1,
%t A207870    {n, 1, 420}]                        (* A207869 *)
%t A207870 Flatten[Position[t1, 0]]               (* this sequence *)
%t A207870 t2 = Table[p[n, x] /. x -> I, {n, 1, 420}];
%t A207870 Flatten[Position[t2, 0]]               (* A207871 *)
%t A207870 Denominator[Table[p[n, x] /. x -> 1/2, {n, 1, 120}]]  (* A207872 *)
%t A207870 Numerator[Table[p[n, x] /. x -> 1/2, {n, 1, 120}]]    (* A207873 *)
%Y A207870 Cf. A207813, A207869.
%K A207870 nonn
%O A207870 1,1
%A A207870 _Clark Kimberling_, Feb 21 2012