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A172197 Decimal expansion of the abscissa x of a local maximum of the Fibonacci function F(x). 1
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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.

LINKS

Table of n, a(n) for n=1..105.

Gerd Lamprecht, Iterationsrechner mit Algorithmus

Gerd Lamprecht, Zahlenfolgen (sequences)

E. Weisstein, Fibonacci Number, Mathworld.

EXAMPLE

F(1.0945761052316...) = 1.0098243...

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-Fran├žois Alcover, Jan 28 2014 *)

PROG

(Other) Gerd Lamprecht online Iterationsrechner: #(@P@C1], x+x)*@C2]+cos(x*PI)*@C2]+sin(x*PI)*PI)*@P@C1], -x)/@C0]@N@C0]=@Q5); @C1]=@C0]/2+0.5; @C2]=log(@C1]); @B1]=1.09; @B2]=1.1; @B3]=Fx(@B1]); @B4]=Fx(@B2]); d=4e-16; IM=2; @N@B4]=Fx(@B2]); @B0]=(@B4]-@B3])/ (@B2]-@B1]); a=@B1]-@B3]/@B0]; b=Fx(a); if(b*@B4]%3C0){@B1]=@B2]; @B2]=a; @B3]=@B4]; }@F@B2]=a; @B3]*=@H2, @B4], b); }@N(@A@B4])%3Cd)@O(@A@B4])%3Cd)@O@A@B2]-@B1])%3Cd@N0@N1@Nif(@A@B4]) %3Cd)c=@B2]; @Eif(@A@B3])%3C1e-16)c=@B1]; @Ec=(@B1]+@B2])/2;

CROSSREFS

Cf. A171909, A172081.

Sequence in context: A240964 A154900 A246546 * A016630 A213614 A099281

Adjacent sequences:  A172194 A172195 A172196 * A172198 A172199 A172200

KEYWORD

cons,nonn

AUTHOR

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

EXTENSIONS

Edited, embedded JavaScript source code of URL removed - R. J. Mathar, Feb 02 2010

STATUS

approved

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Last modified November 14 17:24 EST 2019. Contains 329126 sequences. (Running on oeis4.)