A172197 Decimal expansion of the abscissa x of a local maximum of the Fibonacci function F(x).

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

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-François Alcover, Jan 28 2014 *)

Prog

(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 A374490 A213614

Adjacent sequences: A172194 A172195 A172196 * A172198 A172199 A172200

Keyword

cons,nonn,changed

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