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%I #12 Jan 28 2019 02:54:42
%S 3,0,7,2,5,4,0,4,7,7,6,4,4,8,5,7,5,7,9,0,8,5,9,4,6,5,2,0,8,3,5,4,0,9,
%T 6,5,2,4,4,1,1,2,5,0,0,7,9,1,7,1,1,9,0,0,1,9,1,7,8,2,6,9,5,3,9,3,6,6,
%U 5,0,1,2,3,5,9,0,3,0,5,3,2,4,1,5,5,4,0,0,7,3,7,0,4,3,0,6,2,0,6,8,5,4,8,8,4
%N Decimal expansion of the real part of the continued fraction i/(Pi+i/(Pi+i/(...))).
%C Here, i is the imaginary unit sqrt(-1).
%C The c.f. of which this is the real part converges to one of the two solutions of the equation z*(Pi+z)=i. It is also the unique attractor of the complex mapping M(z)=i/(Pi+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 A263211.
%C Note also that when Pi and i are exchanged, the resulting c.f. Pi/(i+Pi/(i+Pi/(...))) does not converge, and the corresponding mapping has no attractor.
%H Stanislav Sykora, <a href="/A263210/b263210.txt">Table of n, a(n) for n = -1..2000</a>
%F Equals the real part of (sqrt(Pi^2+4*i)-Pi)/2.
%e 0.030725404776448575790859465208354096524411250079171190019178269539...
%t RealDigits[(16 + Pi^4)^(1/4) * Cos[ArcTan[4/Pi^2]/2]/2 - Pi/2, 10, 120][[1]] (* _Vaclav Kotesovec_, Jan 28 2019 *)
%o (PARI) real((-Pi+sqrt(Pi^2+4*I))/2)
%Y Cf. A000796, A263211.
%K nonn,cons
%O -1,1
%A _Stanislav Sykora_, Oct 12 2015
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