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%I #20 Jan 23 2018 06:53:16
%S 4,5,8,2,0,2,3,4,1,3,7,8,3,5,0,2,8,0,6,0,1,5,8,3,1,2,8,8,9,3,5,6,8,8,
%T 8,6,3,6,3,8,3,0,9,6,0,9,5,5,7,8,0,6,1,6,6,3,4,3,5,3,2,7,5,8,1,3,6,7,
%U 5,4,3,7,3,7,6,8,2,3,3,5,0,2,5,6,4,5,6,4,5,5,4,4,7,6,9,2,8,9,6,4,5,6,8,8,1
%N Decimal expansion of the real part of the continued fraction i/(e + i/(e + i/(...))).
%C Here, i is the imaginary unit sqrt(-1) and e is the Euler number.
%C The continued fraction of which this is the real part converges to one of the two solutions of the equation z * (e + z) = i. It is also the unique attractor of the complex mapping M(z) = i/(e + z). The other solution of the equation is an invariant point of M(z), but not its attractor. The imaginary part of this complex constant is in A263209.
%C Note also that when e and i are exchanged, the resulting continued fraction e/(i + e/(i + e/(...))) does not converge, and the corresponding mapping has no attractor.
%H Stanislav Sykora, <a href="/A263208/b263208.txt">Table of n, a(n) for n = -1..2000</a>
%F Equals the real part of (sqrt(e^2 + 4 * i) - e)/2.
%e 0.0458202341378350280601583128893568886363830960955780616634353275813...
%p evalf((16 + exp(4))^(1/4) * cos(arctan(4*exp(-2))/2) / 2 - exp(1)/2, 120); # _Vaclav Kotesovec_, Nov 06 2015
%t RealDigits[Re[(Sqrt[E^2 + 4I] - E)/2], 10, 100][[1]] (* _Alonso del Arte_, Oct 12 2015 *)
%o (PARI) real(-exp(1)+sqrt(exp(2)+4*I))/2
%Y Cf. A001113, A263209.
%K nonn,cons
%O -1,1
%A _Stanislav Sykora_, Oct 12 2015
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