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%I #13 Jun 18 2015 05:57:03
%S 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,
%T 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,
%U 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
%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).
%e 1.300242590220...
%t RealDigits[1/2 + Sqrt[(1+Sqrt[17])/8],10,120][[1]] (* _Vaclav Kotesovec_, May 28 2015 *)
%Y Cf. A156590.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Feb 12 2009
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