A156548 Decimal expansion of the real part of the limit of f(f(...f(0)...)) where f(z)=sqrt(i+z).

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

Offset: 1

Author

Clark Kimberling, Feb 12 2009

Keywords

• nonn
• cons

Comments

The imaginary part, 0.624810..., is given by A156590.

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

Examples

			1.30024259022012041915890982074952138854853281918394761...
		

Crossrefs

Cf. A156590.

Programs

Mathematica

RealDigits[1/2 + Sqrt[(1+Sqrt[17])/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).

Equals real part of 1/2 + Sum_{n>=0} ((-1)^(n/2 + 5/4)*binomial(2*n, n))/(2^(4*n)*(2*n - 1)). - Antonio Graciá Llorente, Nov 20 2024