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A172081 Decimal expansion of the local minimum F(x) of the Fibonacci Function at x = A171909.

%I A172081 #12 Feb 16 2025 08:33:11
%S A172081 8,9,6,9,4,6,3,8,7,4,2,4,6,0,6,1,7,2,9,1,2,6,0,0,3,7,1,0,6,8,7,6,5,4,
%T A172081 4,4,1,7,9,9,9,3,7,5,7,4,2,0,9,1,8,0,5,6,1,6,5,8,2,7,4,6,4,9,6,1,0,3,
%U A172081 8,1,4,1,5,4,0,6,2,4,2,0,8,2,2,4,1,3,4,6,3,5,6,7,1,9,7,5,3,1,4,4,4,7,4,0
%N A172081 Decimal expansion of the local minimum F(x) of the Fibonacci Function at x = A171909.
%C A172081 Define the Fibonacci Function F(x) and its derivative as in A171909.
%C A172081 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 A172081 Set dF(x)/dx = 0 to find the local minimum.
%H A172081 Gerd Lamprecht, <a href="http://www.gerdlamprecht.de/Roemisch_JAVA.htm##@Na=0;@N@B0]=GetKoDezi(-2,0,200);@Ni==0@N1@N0@N#">Iterationsrechner</a>
%H A172081 Gerd Lamprecht, <a href="http://www.gerdlamprecht.de/Zahlenfolgen.html">Zahlenfolgen (sequences)</a>
%H A172081 E. Weisstein, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>, Mathworld.
%e A172081 F(1.67668837258...) = 0.896946387424606172912600371068765...
%p A172081 p := (1+sqrt(5))/2 ; F := (p^x - cos(Pi*x)/p^x )/sqrt(5);
%p A172081 Fpr := diff(F,x) ; Fpr2 := diff(Fpr,x) ;
%p A172081 Digits := 80 ; x0 := 1.67 ;
%p A172081 for n from 1 to 10 do
%p A172081 x0 := evalf(x0-subs(x=x0,Fpr)/subs(x=x0,Fpr2)) ;
%p A172081 print( evalf(subs(x=x0,F))) ;
%p A172081 end do : # _R. J. Mathar_, Feb 02 2010
%t A172081 digits = 104; F[x_] := (GoldenRatio^x - Cos[Pi*x]/GoldenRatio^x)/Sqrt[5]; x0 = x /. FindRoot[F'[x], {x, 2}, WorkingPrecision -> digits+1]; RealDigits[F[x0], 10, digits][[1]] (* _Jean-François Alcover_, Jan 28 2014 *)
%o A172081 (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 A172081 Cf. A171909, A001622, A030171, A172197.
%K A172081 cons,nonn
%O A172081 0,1
%A A172081 Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Jan 25 2010
%E A172081 Edited, offset and leading zero normalized by _R. J. Mathar_, Feb 02 2010