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A172081 Decimal expansion of the local minimum F(x) of the Fibonacci Function at x = A171909. 4
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 (list; constant; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Jan 28 2014 *)

PROG

(Other) 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: A203146 A203128 A178856 * A154842 A195057 A021531

Adjacent sequences:  A172078 A172079 A172080 * A172082 A172083 A172084

KEYWORD

cons,nonn

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

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Last modified May 26 09:12 EDT 2017. Contains 287093 sequences.