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A207871 Numbers matched to Zeckendorf polynomials divisible by x^2 + 1.

%I A207871 #9 Jun 27 2025 00:45:22
%S A207871 4,7,11,18,22,29,33,36,47,51,54,58,76,80,83,87,94,116,123,127,130,134,
%T A207871 141,145,152,156,163,174,188,192,199,203,206,210,217,221,228,232,235,
%U A207871 246,250,253,264,282,304,311,322,326,329,333,340,344,351,355
%N A207871 Numbers matched to Zeckendorf polynomials divisible by x^2 + 1.
%C A207871 The Zeckendorf polynomials Z(x,n) are defined and ordered at A207813.
%t A207871 fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]],
%t A207871    t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k],
%t A207871     AppendTo[fr, 1]; t = t - Fibonacci[k],
%t A207871     AppendTo[fr, 0]]; k--]; fr]; t = Table[fb[n],
%t A207871       {n, 1, 500}];
%t A207871 b[n_] := Reverse[Table[x^k, {k, 0, n}]]
%t A207871 p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
%t A207871 Table[p[n, x], {n, 1, 40}]
%t A207871 t2 = Table[p[n, x] /. x -> I, {n, 1, 420}];
%t A207871 Flatten[Position[t2, 0]]                              (* A207871 *)
%t A207871 Denominator[Table[p[n, x] /. x -> 1/2, {n, 1, 120}]]  (* A207872 *)
%t A207871 Numerator[Table[p[n, x] /. x -> 1/2, {n, 1, 120}]]    (* A207873 *)
%Y A207871 Cf. A207813.
%K A207871 nonn
%O A207871 1,1
%A A207871 _Clark Kimberling_, Feb 21 2012