A263210 Decimal expansion of the real part of the continued fraction i/(Pi+i/(Pi+i/(...))).

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, 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, 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

Offset

-1,1

Comments

Here, i is the imaginary unit sqrt(-1).

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.

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.

Links

Stanislav Sykora, Table of n, a(n) for n = -1..2000

Formula

Equals the real part of (sqrt(Pi^2+4*i)-Pi)/2.

Example

0.030725404776448575790859465208354096524411250079171190019178269539...

Mathematica

RealDigits[(16 + Pi^4)^(1/4) * Cos[ArcTan[4/Pi^2]/2]/2 - Pi/2, 10, 120][[1]] (* Vaclav Kotesovec, Jan 28 2019 *)

Prog

(PARI) real((-Pi+sqrt(Pi^2+4*I))/2)

Crossrefs

Cf. A000796, A263211.

Sequence in context: A010600 A175918 A265205 * A298095 A327803 A199667

Adjacent sequences: A263207 A263208 A263209 * A263211 A263212 A263213

Keyword

nonn,cons

Author

Stanislav Sykora, Oct 12 2015

Status

approved