A263208 Decimal expansion of the real part of the continued fraction i/(e + i/(e + i/(...))).

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, 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, 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

Offset

-1,1

Comments

Here, i is the imaginary unit sqrt(-1) and e is the Euler number.

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.

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.

Links

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

Formula

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

Example

0.0458202341378350280601583128893568886363830960955780616634353275813...

Maple

evalf((16 + exp(4))^(1/4) * cos(arctan(4*exp(-2))/2) / 2 - exp(1)/2, 120); # Vaclav Kotesovec, Nov 06 2015

Mathematica

RealDigits[Re[(Sqrt[E^2 + 4I] - E)/2], 10, 100][[1]] (* Alonso del Arte, Oct 12 2015 *)

Prog

(PARI) real(-exp(1)+sqrt(exp(2)+4*I))/2

Crossrefs

Cf. A001113, A263209.

Sequence in context: A082468 A152974 A180307 * A010477 A195386 A259468

Adjacent sequences: A263205 A263206 A263207 * A263209 A263210 A263211

Keyword

nonn,cons

Author

Stanislav Sykora, Oct 12 2015

Status

approved