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

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

1,2

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-François Alcover, Feb 22 2013 *)

Prog

(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: A261024 A343436 A186282 * A199079 A010723 A244588

Adjacent sequences: A171906 A171907 A171908 * A171910 A171911 A171912

Keyword

cons,nonn,changed

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