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

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

Author

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

Keywords

• cons
• nonn

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

• Gerd Lamprecht, Iterationsrechner mit Algorithmus
• Gerd Lamprecht, Zahlenfolgen (sequences)
• E. Weisstein, Fibonacci Number, Mathworld.

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