A156590 Decimal expansion of the imaginary part of the limit of f(f(...f(0)...)) where f(z)=sqrt(i+z).

6, 2, 4, 8, 1, 0, 5, 3, 3, 8, 4, 3, 8, 2, 6, 5, 8, 6, 8, 7, 9, 6, 0, 4, 4, 4, 7, 4, 4, 2, 8, 5, 1, 4, 4, 4, 0, 0, 5, 2, 3, 4, 4, 5, 6, 4, 1, 9, 0, 0, 2, 3, 2, 7, 4, 7, 0, 1, 5, 4, 3, 1, 4, 6, 5, 3, 1, 7, 1, 0, 5, 5, 4, 3, 9, 4, 9, 6, 4, 0, 7, 0, 5, 2, 4, 5, 2, 8, 9, 1, 2, 7, 5, 5, 3, 2, 9, 5, 0, 9, 1, 7, 3, 1, 7

Offset: 0

Author

Clark Kimberling, Feb 12 2009

Keywords

• nonn
• cons

Comments

The real part, 1.300242590..., is given by A156548.

Coincides with the limit of the imaginary part of the same expression, but with f(z)=i/(1+z), and therefore with the imaginary part of the continued fraction i/(1+i/(1+i/(...))). It is also equal to the real part of the continued fraction i/(i+i/(i+i/(...))). - Stanislav Sykora, May 27 2015

Examples

			0.6248105338...
		

Crossrefs

Cf. A156548.

Programs

Mathematica

RealDigits[Sqrt[(Sqrt[17]-1)/8],10,120][[1]] (* Vaclav Kotesovec, May 28 2015 *)

Formula

Define z(1)=f(0)=sqrt(i), where i=sqrt(-1), and z(n)=f(z(n-1)) for n>1.

Write the limit of z(n) as a+bi where a and b are real. Then a=(b+1)/(2b), where b=sqrt((sqrt(17)-1)/8).