A207871 -id:A207871 - cp's OEIS frontend

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

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, 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, 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

Offset: 1

Clark Kimberling, Feb 21 2012

sign

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

			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.

Cf. A207813, A207870.

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}]
Table[p[n, x] /. x -> 1, {n, 1, 120}]  (* A007895 *)
Table[p[n, x] /. x -> 2, {n, 1, 120}]  (* A003714 *)
Table[p[n, x] /. x -> 3, {n, 1, 120}]  (* A060140 *)
   t1 = Table[p[n, x] /. x -> -1,
   {n, 1, 420}]                        (* A207869 *)
Flatten[Position[t1, 0]]               (* A207870 *)
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 *)

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

Original entry on oeis.org

6, 10, 14, 16, 23, 26, 35, 37, 42, 51, 57, 60, 68, 74, 83, 90, 92, 97, 106, 110, 116, 120, 127, 132, 134, 146, 149, 157, 163, 172, 178, 184, 188, 192, 194, 206, 214, 217, 234, 236, 241, 250, 254, 260, 264, 271, 276, 278, 288, 294, 298, 302, 304, 311

Offset: 1

Clark Kimberling, Feb 21 2012

nonn

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

			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.

Cf. A207813, A207869.

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}]
Table[p[n, x] /. x -> 1, {n, 1, 120}]  (* A007895 *)
Table[p[n, x] /. x -> 2, {n, 1, 120}]  (* A003714 *)
Table[p[n, x] /. x -> 3, {n, 1, 120}]  (* A060140 *)
   t1 = Table[p[n, x] /. x -> -1,
   {n, 1, 420}]                        (* A207869 *)
Flatten[Position[t1, 0]]               (* this sequence *)
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 *)

cp's OEIS Frontend

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

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