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%I #11 Feb 22 2013 09:56:50
%S 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,
%T 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,
%U 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
%N Decimal expansion of the abscissa x of a local minimum of the Fibonacci Function F(x).
%C 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.
%C The derivative is dF/dx = ( phi^x * log(phi) - cos(Pi*x) *log(phi)/ phi^x + Pi*sin(Pi*x)/ phi^x)/sqrt(5).
%C Set dF(x)/dx=0 to find local minima and maxima.
%H Gerd Lamprecht, <a href="http://www.gerdlamprecht.de/Roemisch_JAVA.htm#ZZZZZ0058">Iterationsrechner mit Algorithmus</a>
%H Gerd Lamprecht, <a href="http://www.gerdlamprecht.de/Zahlenfolgen.html">Zahlenfolgen (sequence)</a>
%H E. Weisstein, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>, Mathworld.
%H A. Stakhov and Boris Rozin, <a href="http://dx.doi.org/10.1016/j.chaos.2005.08.158">The continuous functions for the Fibonacci and Lucas p-numbers</a>, Chaos, Solit. and Fractals 28 (4) (2006) 1014-1025.
%e F(1.67668837258...)=0.896946387424606172912600371068765... = A172081
%t 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 *)
%o (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);
%Y Cf. A001622, A089260
%K cons,nonn
%O 1,2
%A Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Dec 31 2009
%E Description edited, JavaScript calculations embedded in URL's removed, Weisstein and Stakhov-Rozin ref added by _R. J. Mathar_, Feb 02 2010
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