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%I #11 Aug 01 2022 16:23:11
%S 3,3,0,1,1,4,2,1,4,8,5,2,8,7,0,2,0,2,8,8,9,3,2,9,5,8,8,7,7,2,2,8,2,6,
%T 8,2,5,7,3,6,9,8,5,0,0,8,3,2,6,3,7,6,3,8,7,8,1,9,6,0,0,2,4,5,1,9,3,5,
%U 9,1,5,2,7,5,6,1,6,5,6,9,8,3,7,2,6,6,8,5,0,4,2,4,0,4,4,2,0,6,3,6,7,6,4,6
%N Decimal expansion of solution to x=Fibonacci(x) with 0<x<1.
%C Fixed point of the Fibonacci function defined as F(x) = ( phi^x - cos(Pi*x) / phi^x )/sqrt(5), an interpolation of the Fibonacci numbers, with phi = A001622, Pi = A000796.
%H Gerd Lamprecht, <a href="http://www.gerdlamprecht.de/Roemisch_JAVA.htm#@P@Q5)*0.5+0.5,x)/@Q5)+@P@Q5)*0.5-0.5,x)*sin(PI*(x-0.5))/@Q5)-x@Na=0.33;b=0.331;c=(a+b)/2;@Nd=(Fx(c)*Fx(a)%3C0);a=d?a:c;b=d?c:b;c=(c+(d?a:b))/2;@N@AFx(c))%3C%205e-17@N0@N1@Nc=c;@B0]=GetKoDezi(-11,0,56);">Iterationsrechner mit Algorithmus</a>
%H Gerd Lamprecht, <a href="http://www.gerdlamprecht.de/Zahlenfolgen.html">Zahlenfolgen (sequence)</a>
%F Gerd Lamprecht online Iterationsrechner Beispiel 59.
%e 0.3301142148528702028... = Fibonacci(0.3301142148528702028...)
%t RealDigits[x/.FindRoot[x==Fibonacci[x],{x,.3},WorkingPrecision->120]] [[1]] (* _Harvey P. Dale_, Jan 19 2015 *)
%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)-x@Na=0.33;b=0.331;c=(a+b)/2;@Nd=(Fx(c)*Fx(a)%3C0);a=d?a:c;b=d?c:b;c=(c+(d?a:b))/2;@N@AFx(c))%3C%205e-17@N0@N1@Nc=c;@B0]=GetKoDezi(-11,0,56);
%o (PARI) F(x) = my(phi=(sqrt(5)+1)/2); (phi^x - cos(Pi*x)/phi^x)/sqrt(5);
%o solve(x=0.2, 0.8, x-F(x)) \\ _Michel Marcus_, Jul 29 2022
%Y Cf. A171909, A172081.
%K cons,nonn
%O 0,1
%A Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Jan 28 2010
%E Adjusted offset and leading zero from _R. J. Mathar_, Jan 30 2010
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