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A172169
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Decimal expansion of solution to x=Fibonacci(x) with 0<x<1.
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0
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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, 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, 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
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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Gerd Lamprecht online Iterationsrechner Beispiel 59.
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EXAMPLE
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0.3301142148528702028... = Fibonacci(0.3301142148528702028...)
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MATHEMATICA
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RealDigits[x/.FindRoot[x==Fibonacci[x], {x, .3}, WorkingPrecision->120]] [[1]] (* Harvey P. Dale, Jan 19 2015 *)
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PROG
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(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);
(PARI) F(x) = my(phi=(sqrt(5)+1)/2); (phi^x - cos(Pi*x)/phi^x)/sqrt(5);
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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 28 2010
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EXTENSIONS
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Adjusted offset and leading zero from R. J. Mathar, Jan 30 2010
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STATUS
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approved
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