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 1-4 of 4 results.

A192038 Decimal approximation of x such that f(x)=4, where f is the Fibonacci function.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jun 21 2011

Keywords

Examples

			4.549112556507743239203225039690296777977751571212553...

Links

Crossrefs

Cf. A192039, A192040, A192041, A192042, A192043, A192044 (these correspond to f(x) = 6, 7, 1/2, 3/2, phi, phi^2 respectively); A171909, A172081.

Programs

  • Mathematica
    r = GoldenRatio; s = 1/Sqrt[5];
f[x_] := s*(r^x - Cos[Pi*x] * r^(-x));
x /. FindRoot[Fibonacci[x] == 4, {x, 5}, WorkingPrecision -> 100]
RealDigits[%, 10]
(Show[Plot[#1, #2], ListPlot[Table[{x, #1}, #2]]] &)[
Fibonacci[x], {x, -7, 7}] (* Peter J. C. Moses, Jun 21 2011 *)
  • PARI
    phi = (1+sqrt(5))/2; solve(x=4, 5, (phi^x - cos(Pi*x) * phi^(-x))/sqrt(5) - 4) \\ Michel Marcus, Oct 05 2016

Formula

f(x) = (phi^x - cos(Pi*x) * phi^(-x))/sqrt(5), where phi = (1+sqrt(5))/2 (the golden ratio). The function f, a generalization over the reals of the Binet formula, gives Fibonacci numbers for integer values of x; e.g., f(3) = 2, f(4) = 3, f(5) = 5. [Corrected by Daniel Forgues, Oct 05 2016]

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

A172197 Decimal expansion of the abscissa x of a local maximum of the Fibonacci function F(x).

Original entry on oeis.org

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

Views

Author

Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Jan 29 2010

Keywords

Comments

Define the Fibonacci Function F(x) and its derivative dF/dx as in A172081.
At the local maximum, dF(x)/dx = 0.
This constant x=1.0945... here satisfies this condition of vanishing first derivative.

Examples

			F(1.0945761052316...) = 1.0098243...

Links

Crossrefs

Cf. A171909, A172081.

Programs

  • Maple
    p := (1+sqrt(5))/2 ; F := (p^x - cos(Pi*x)/p^x )/sqrt(5);
Fpr := diff(F,x) ; Fpr2 := diff(Fpr,x) ;
Digits := 80 ; x0 := 1.0 ;
for n from 1 to 10 do
x0 := evalf(x0-subs(x=x0,Fpr)/subs(x=x0,Fpr2)) ;
end do ; # R. J. Mathar, Feb 02 2010
  • Mathematica
    digits = 105; F[x_] := (GoldenRatio^x - Cos[Pi*x]/GoldenRatio^x)/Sqrt[5]; x0 = x /. FindRoot[F'[x], {x, 1}, WorkingPrecision -> digits+1]; RealDigits[x0, 10, digits][[1]] (* Jean-François Alcover, Jan 28 2014 *)

Extensions

Edited, embedded JavaScript source code of URL removed - R. J. Mathar, Feb 02 2010

A172169 Decimal expansion of solution to x=Fibonacci(x) with 0

Original entry on oeis.org

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

Views

Author

Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Jan 28 2010

Keywords

Comments

Fixed point of the Fibonacci function defined as F(x) = ( phi^x - cos(Pi*x) / phi^x )/sqrt(5), an interpolation of the Fibonacci numbers, with phi = A001622, Pi = A000796.

Examples

			0.3301142148528702028... = Fibonacci(0.3301142148528702028...)

Links

Crossrefs

Cf. A171909, A172081.

Programs

  • Mathematica
    RealDigits[x/.FindRoot[x==Fibonacci[x],{x,.3},WorkingPrecision->120]] [[1]] (* Harvey P. Dale, Jan 19 2015 *)
  • PARI
    F(x) = my(phi=(sqrt(5)+1)/2); (phi^x - cos(Pi*x)/phi^x)/sqrt(5);
solve(x=0.2, 0.8, x-F(x)) \\ Michel Marcus, Jul 29 2022

Formula

Gerd Lamprecht online Iterationsrechner Beispiel 59.

Extensions

Adjusted offset and leading zero from R. J. Mathar, Jan 30 2010
Showing 1-4 of 4 results.

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