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 1-3 of 3 results.

A171909 Decimal expansion of the abscissa x of a local minimum of the Fibonacci Function F(x).

Original entry on oeis.org

1, 6, 7, 6, 6, 8, 8, 3, 7, 2, 5, 8, 1, 5, 8, 4, 1, 9, 2, 6, 2, 3, 3, 8, 4, 7, 4, 4, 6, 1, 6, 0, 2, 6, 0, 7, 7, 8, 5, 9, 0, 8, 9, 3, 4, 0, 6, 1, 1, 7, 5, 2, 0, 3, 4, 7, 5, 1, 6, 5, 6, 5, 0, 6, 5, 2, 5, 0, 3, 2, 1, 0, 4, 8, 9, 6, 8, 1, 5, 8, 2, 1, 5, 7, 8, 9, 7, 9, 2, 4, 9, 6, 6, 9, 8, 0, 7, 5, 9, 5, 0, 1, 5, 7, 4
Offset: 1

Views

Author

Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Dec 31 2009

Keywords

Comments

Define the Fibonacci Function F(x) = ( phi^x - cos(Pi*x) / phi^x )/sqrt(5) as an interpolation of the Fibonacci numbers, with phi = A001622, Pi = A000796.
The derivative is dF/dx = ( phi^x * log(phi) - cos(Pi*x) *log(phi)/ phi^x + Pi*sin(Pi*x)/ phi^x)/sqrt(5).
Set dF(x)/dx=0 to find local minima and maxima.

Examples

			F(1.67668837258...)=0.896946387424606172912600371068765... = A172081

Links

Crossrefs

Cf. A001622, A089260

Programs

  • Mathematica
    x /. FindRoot[((1 + Sqrt[5])/2)^(2*x)*ArcCsch[2] + ArcCsch[2]*Cos[Pi*x] + Pi*Sin[Pi*x], {x, 2}, WorkingPrecision -> 105] // RealDigits // First (* Jean-François Alcover, Feb 22 2013 *)

Extensions

Description edited, JavaScript calculations embedded in URL's removed, Weisstein and Stakhov-Rozin ref added by R. J. Mathar, Feb 02 2010

A353920 Decimal expansion of the first positive real root of ((1 - sqrt(5))*((1 + sqrt(5)) /2)^x - (1 + sqrt(5))*((1 - sqrt(5))/2)^x)/(2*sqrt(5)).

Original entry on oeis.org

8, 1, 6, 1, 9, 7, 6, 4, 0, 3, 0, 7, 0, 4, 4, 3, 9, 5, 0, 8, 6, 0, 3, 0, 9, 8, 9, 8, 4, 8, 7, 3, 3, 2, 6, 5, 7, 4, 2, 8, 7, 7, 2, 8, 0, 1, 3, 4, 6, 5, 7, 1, 8, 2, 9, 0, 5, 0, 3, 9, 1, 7, 2, 2, 9, 8, 5, 5, 2, 1, 0, 5, 9, 5, 2, 2, 5, 9, 3, 8, 5, 4, 3, 3, 4, 5, 0, 3, 6, 5, 1, 4, 1, 2, 1, 6, 2, 6, 6, 0, 3, 8, 5, 8, 2
Offset: 0

Views

Author

Peter Luschny, May 11 2022

Keywords

Comments

The constant is the abscissa of the first intercept point of the row functions for x > 0 of the generalized Fibonacci function A353595, see illustration.

Examples

			0.816197640307044395086030989848733265742877280134657182905...

Links

Crossrefs

Cf. A001622, A353595, A089260.

Programs

  • Maple
    sphi := x -> ((1-sqrt(5))*((1+sqrt(5))/2)^x - (1 + sqrt(5))*((1 - sqrt(5))/2)^x)/ (2*sqrt(5)):
Digits := 120: fsolve(Re(sphi(x)) = 0, x, 0.7..0.9, fulldigits)*10^105:
ListTools:-Reverse(convert(floor(%), base, 10));
  • Mathematica
    sphi[x_] := 2 Re[ Exp[-I Pi x / 2] Sin[(x - 1)(Pi / 2 - I ArcCsch[2])]] / Sqrt[5];
x /. FindRoot[Sphi[x], {x, 0.8}, WorkingPrecision -> 120]
RealDigits[%, 10, 105][[1]]

Formula

Equals the first positive real root of 2*exp(-I*Pi*x/2)*sin((x - 1)*(Pi/2 - I * arccsch(2))) / sqrt(5).

A089261 Continued fraction expansion of -x, the largest negative root of the equation Fibonacci(x) = 0.

Original entry on oeis.org

0, 5, 2, 3, 1, 2, 2, 5, 6, 13, 2, 7, 2, 2, 1, 1, 5, 1, 4, 1, 4, 16, 1, 1, 2, 1, 1, 4, 2, 1, 1, 3, 1, 2, 2, 1, 6, 1, 4, 4, 1, 7, 2, 1, 2, 3, 1, 1, 7, 2, 1, 16, 1, 2, 1, 1, 67, 462, 1, 2, 4, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 14, 1, 1, 2, 1, 1, 1, 9, 2, 2, 1, 1107, 2, 3, 10, 2, 4, 1, 1, 1, 1, 1, 44, 1, 1, 4
Offset: 0

Views

Author

T. D. Noe, Oct 27 2003

Keywords

Links

Crossrefs

Cf. A089260 (decimal expansion).

Programs

  • Mathematica
    ContinuedFraction[ -x/.FindRoot[Fibonacci[x]==0, {x, -0.2}, WorkingPrecision->100], 100]

Extensions

Offset changed by Andrew Howroyd, Aug 07 2024
Showing 1-3 of 3 results.

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