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A172197
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Decimal expansion of the abscissa x of a local maximum of the Fibonacci function F(x).
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1
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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F(1.0945761052316...) = 1.0098243...
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MAPLE
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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)) ;
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MATHEMATICA
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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 *)
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PROG
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(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;
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CROSSREFS
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KEYWORD
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AUTHOR
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Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Jan 29 2010
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EXTENSIONS
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Edited, embedded JavaScript source code of URL removed - R. J. Mathar, Feb 02 2010
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STATUS
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approved
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