cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A207872 Denominator of Z(n,1/2), where Z(n,x) is the n-th Zeckendorf polynomial.

Original entry on oeis.org

1, 2, 4, 4, 8, 8, 8, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 256, 256, 256, 256, 256
Offset: 1

Views

Author

Clark Kimberling, Feb 21 2012

Keywords

Comments

The Zeckendorf polynomials Z(x,n) are defined and ordered at A207813. Each power 2^k appears F(k+1) times, where F=A000045 (Fibonacci numbers).
Conjecture: a(n) is also the reverse binarization of the Zeckendorf representation of n in base Fibonacci. For example, 11 = 1x8 + 0x5 +1x3 +0x2 + 0x1, so 11 =10100 in base Fibonacci. Now read that as binary but in reverse, 00101 = 101 = 5 = A207873(11). - George Beck, Sep 02 2020

Links

Crossrefs

Cf. A207813, A207873.

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}]
Denominator[Table[p[n, x] /. x -> 1/2,
   {n, 1, 120}]]                       (* A207872 *)
Numerator[Table[p[n, x] /. x -> 1/2,
   {n, 1, 120}]]                       (* A207873 *)

A207873 Numerator of Z(n,1/2), where Z(n,x) is the n-th Zeckendorf polynomial.

Original entry on oeis.org

1, 1, 1, 5, 1, 9, 5, 1, 17, 9, 5, 21, 1, 33, 17, 9, 41, 5, 37, 21, 1, 65, 33, 17, 81, 9, 73, 41, 5, 69, 37, 21, 85, 1, 129, 65, 33, 161, 17, 145, 81, 9, 137, 73, 41, 169, 5, 133, 69, 37, 165, 21, 149, 85, 1, 257, 129, 65, 321, 33, 289, 161, 17, 273, 145, 81, 337
Offset: 1

Views

Author

Clark Kimberling, Feb 21 2012

Keywords

Comments

The Zeckendorf polynomials Z(x,n) are defined and ordered at A207813. See A207872 for denominators to match A207873.

Crossrefs

Cf. A207813, A207873.

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}]
Denominator[Table[p[n, x] /. x -> 1/2,
   {n, 1, 120}]]                       (* A207872 *)
Numerator[Table[p[n, x] /. x -> 1/2,
   {n, 1, 120}]]                       (* A207873 *)

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 21 2012

Keywords

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 *)

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 21 2012

Keywords

Comments

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

Examples

			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.

Crossrefs

Cf. A207813, A207870.

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}]
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

Views

Author

Clark Kimberling, Feb 21 2012

Keywords

Comments

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

Examples

			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.

Crossrefs

Cf. A207813, A207869.

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}]
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 *)

A207822 Number of distinct irreducible factors of n-th Zeckendorf polynomial.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 2, 3, 1, 2, 1, 3, 1, 2, 3, 2, 1, 3, 1, 2, 2, 1, 2, 3, 2, 1, 2, 3, 3, 2, 2, 1, 2, 3, 1, 2, 2, 1, 2, 2, 2, 2, 3, 3, 1, 3, 1, 1, 3, 3, 1, 3, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 2, 2, 1, 3, 2, 2, 2, 1, 4, 3, 1, 2, 3, 2, 1, 3, 2, 3, 1, 3, 1, 2, 3, 1, 2
Offset: 1

Views

Author

Clark Kimberling, Feb 20 2012

Keywords

Comments

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

Examples

			Z(10,n) = x^4 + x = x(x + 1)(x^2 - x + 1), so a(10)=3.

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]]]]
TableForm[Table[{n, p[n, x], FactorList[p[n, x]]},
{n, 1, 10}]]
Table[-1 + Length[FactorList[p[n, x]]], {n, 1, 120}]
Showing 1-6 of 6 results.

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