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A171909 Decimal expansion of the abscissa x of a local minimum of the Fibonacci Function F(x). 4
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..105.

Gerd Lamprecht, Iterationsrechner mit Algorithmus

Gerd Lamprecht, Zahlenfolgen (sequence)

E. Weisstein, Fibonacci Number, Mathworld.

A. Stakhov and Boris Rozin, The continuous functions for the Fibonacci and Lucas p-numbers, Chaos, Solit. and Fractals 28 (4) (2006) 1014-1025.

EXAMPLE

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

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-Fran├žois Alcover, Feb 22 2013 *)

PROG

(Other) Gerd Lamprecht online Iterationsrechner: #@P@Q5)*0.5+0.5, x)/@Q5)+@P@Q5)*0.5-0.5, x)*sin(PI*(x-0.5))/@Q5)@Na=0.19; b=1.6; @B2]=2; @N@B0]=Fx(b); @B1]=Fx(b-a); @B2]=Fx(b+a); if(@B0]%3C@B1]&&@B0]%3C@B2])a/=10; @Eif(@B1]%3C@B2])b-=a; @Eb+=a; @N@A@B1]-@B2])%3C1e-17@N1@N1@Nc=Fx(b);

CROSSREFS

Cf. A001622, A089260

Sequence in context: A120962 A261024 A186282 * A199079 A010723 A244588

Adjacent sequences:  A171906 A171907 A171908 * A171910 A171911 A171912

KEYWORD

cons,nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified June 5 09:01 EDT 2020. Contains 334829 sequences. (Running on oeis4.)