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 A171909 Decimal expansion of the abscissa x of a local minimum of the Fibonacci Function F(x). 4
 1, 6, 7, 6, 6, 8, 8, 3, 7, 2, 5, 8, 1, 5, 8, 4, 1, 9, 2, 6, 2, 3, 3, 8, 4, 7, 4, 4, 6, 1, 6, 0, 2, 6, 0, 7, 7, 8, 5, 9, 0, 8, 9, 3, 4, 0, 6, 1, 1, 7, 5, 2, 0, 3, 4, 7, 5, 1, 6, 5, 6, 5, 0, 6, 5, 2, 5, 0, 3, 2, 1, 0, 4, 8, 9, 6, 8, 1, 5, 8, 2, 1, 5, 7, 8, 9, 7, 9, 2, 4, 9, 6, 6, 9, 8, 0, 7, 5, 9, 5, 0, 1, 5, 7, 4 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Define the Fibonacci Function F(x) = ( phi^x - cos(Pi*x) / phi^x )/sqrt(5) as an interpolation of the Fibonacci numbers, with phi = A001622, Pi = A000796. 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 local minima and maxima. LINKS Gerd Lamprecht, Iterationsrechner mit Algorithmus Gerd Lamprecht, Zahlenfolgen (sequence) E. Weisstein, Fibonacci Number, Mathworld. A. Stakhov and Boris Rozin, The continuous functions for the Fibonacci and Lucas p-numbers, Chaos, Solit. and Fractals 28 (4) (2006) 1014-1025. EXAMPLE F(1.67668837258...)=0.896946387424606172912600371068765... = A172081 MATHEMATICA x /. FindRoot[((1 + Sqrt[5])/2)^(2*x)*ArcCsch[2] + ArcCsch[2]*Cos[Pi*x] + Pi*Sin[Pi*x], {x, 2}, WorkingPrecision -> 105] // RealDigits // First (* Jean-François Alcover, Feb 22 2013 *) 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. A001622, A089260 Sequence in context: A120962 A261024 A186282 * A199079 A010723 A244588 Adjacent sequences:  A171906 A171907 A171908 * A171910 A171911 A171912 KEYWORD cons,nonn AUTHOR Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Dec 31 2009 EXTENSIONS Description edited, JavaScript calculations embedded in URL's removed, Weisstein and Stakhov-Rozin ref added by R. J. Mathar, Feb 02 2010 STATUS approved

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Last modified June 5 09:01 EDT 2020. Contains 334829 sequences. (Running on oeis4.)