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

Author

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

Keywords

• cons
• nonn

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

• 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.

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