%N Decimal expansion of the real part of the limit of f(f(...f(0)...)) where f(z)=sqrt(i+z).
%C The imaginary part, 0.624810..., is given by A156590.
%C (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
%F Define z(1)=f(0)=sqrt(i), where i=sqrt(-1), and z(n)=f(z(n-1)) for n>1.
%F 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).
%t RealDigits[1/2 + Sqrt[(1+Sqrt)/8],10,120][] (* _Vaclav Kotesovec_, May 28 2015 *)
%Y Cf. A156590.
%A _Clark Kimberling_, Feb 12 2009