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

0,1

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.

Links

Table of n, a(n) for n=0..103.

Gerd Lamprecht, Iterationsrechner

Gerd Lamprecht, Zahlenfolgen (sequences)

E. Weisstein, Fibonacci Number, Mathworld.

Example

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

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 *)

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. A171909, A001622, A030171, A172197.

Sequence in context: A203128 A178856 A321161 * A154842 A195057 A021531

Adjacent sequences: A172078 A172079 A172080 * A172082 A172083 A172084

Keyword

cons,nonn,changed

Author

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

Extensions

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

Status

approved