A172081 Decimal expansion of the local minimum F(x) of the Fibonacci Function at x = A171909.

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

Offset: 0

Author

Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Jan 25 2010

Keywords

• cons
• nonn

Comments

Define the Fibonacci Function F(x) and its derivative as in A171909.

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 the local minimum.

Examples

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

Links

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

Crossrefs

Cf. A171909, A001622, A030171, A172197.

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 := 1.67 ;
for n from 1 to 10 do
x0 := evalf(x0-subs(x=x0,Fpr)/subs(x=x0,Fpr2)) ;
print( evalf(subs(x=x0,F))) ;
end do : # R. J. Mathar, Feb 02 2010

Mathematica

digits = 104; F[x_] := (GoldenRatio^x - Cos[Pi*x]/GoldenRatio^x)/Sqrt[5]; x0 = x /. FindRoot[F'[x], {x, 2}, WorkingPrecision -> digits+1]; RealDigits[F[x0], 10, digits][[1]] (* Jean-François Alcover, Jan 28 2014 *)

Extensions

Edited, offset and leading zero normalized by R. J. Mathar, Feb 02 2010