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

4, 7, 11, 18, 22, 29, 33, 36, 47, 51, 54, 58, 76, 80, 83, 87, 94, 116, 123, 127, 130, 134, 141, 145, 152, 156, 163, 174, 188, 192, 199, 203, 206, 210, 217, 221, 228, 232, 235, 246, 250, 253, 264, 282, 304, 311, 322, 326, 329, 333, 340, 344, 351, 355

Offset: 1

Author

Clark Kimberling, Feb 21 2012

Keywords

• nonn

Comments

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

Crossrefs

Cf. A207813.

Programs

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}]
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 *)