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

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

Author

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

Keywords

• cons
• nonn

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.

Examples

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

Links

• Gerd Lamprecht, Iterationsrechner mit Algorithmus
• Gerd Lamprecht, Zahlenfolgen (sequence)

Crossrefs

Cf. A171909, A172081.

Programs

Mathematica

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

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

Formula

Gerd Lamprecht online Iterationsrechner Beispiel 59.

Extensions

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