A172169 Decimal expansion of solution to x=Fibonacci(x) with 0<x<1.

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

Offset

0,1

Comments

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.

Links

Table of n, a(n) for n=0..103.

Gerd Lamprecht, Iterationsrechner mit Algorithmus

Gerd Lamprecht, Zahlenfolgen (sequence)

Formula

Gerd Lamprecht online Iterationsrechner Beispiel 59.

Example

0.3301142148528702028... = Fibonacci(0.3301142148528702028...)

Mathematica

RealDigits[x/.FindRoot[x==Fibonacci[x], {x, .3}, WorkingPrecision->120]] [[1]] (* Harvey P. Dale, Jan 19 2015 *)

Prog

(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);
solve(x=0.2, 0.8, x-F(x)) \\ Michel Marcus, Jul 29 2022

Crossrefs

Cf. A171909, A172081.

Sequence in context: A163535 A288395 A288644 * A306629 A185282 A193470

Adjacent sequences: A172166 A172167 A172168 * A172170 A172171 A172172

Keyword

cons,nonn,changed

Author

Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Jan 28 2010

Extensions

Adjusted offset and leading zero from R. J. Mathar, Jan 30 2010

Status

approved