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A272875 Decimal expansion of the real part of the infinite nested power (1+(1+(1+...)^i)^i)^i, with i being the imaginary unit. 3
6, 7, 3, 8, 8, 1, 3, 3, 1, 1, 0, 7, 8, 7, 5, 5, 1, 5, 7, 8, 0, 2, 3, 1, 1, 9, 0, 4, 6, 8, 1, 0, 1, 9, 3, 3, 8, 7, 6, 4, 5, 0, 3, 3, 4, 7, 9, 3, 3, 7, 2, 5, 4, 5, 4, 8, 9, 9, 8, 1, 3, 5, 1, 6, 6, 9, 3, 1, 2, 1, 3, 0, 3, 2, 7, 5, 2, 9, 3, 8, 0, 7, 5, 0, 8, 5, 0, 7, 6, 3, 1, 1, 7, 3, 1, 7, 9, 4, 1, 6, 3, 6, 1, 0, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The mapping M(z)=(1+z)^i has in C a unique invariant point, namely z0 = a+A272876*i, which is also its attractor. Iterative applications of M applied to any starting complex point z (except for the singular value -1+0*i) rapidly converge to z0. The convergence, and the existence of this limit, justify the expression used in the name. It is easy to show that, close to z0, the convergence is exponential, with the error decreasing approximately by a factor of abs(z0/(1+z0))=0.4571... per iteration.

The imaginary part and the modulus of this complex constant are in A272876 and A272877, respectively.

LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..2000

FORMULA

z0 = a+A272876*i satisfies the equations (1+z0)^i = z0, (1+z0)*z0^i = 1.

EXAMPLE

0.6738813311078755157802311904681019338764503347933725454899813516...

PROG

(PARI) \\ f(x) computes (x+(x+...)^i)^i, provided that it converges:

f(x)={my(z=1.0, zlast=0.0, eps=10.0^(1-default(realprecision))); while(abs(z-zlast)>eps, zlast=z; z=(x+z)^I); return(z)}

\\ To compute this constant, use:

z0 = f(1); real(z0)

CROSSREFS

Cf. A156548, A272876, A272877.

Sequence in context: A197141 A139350 A092560 * A018248 A146485 A049254

Adjacent sequences:  A272872 A272873 A272874 * A272876 A272877 A272878

KEYWORD

nonn,cons

AUTHOR

Stanislav Sykora, May 15 2016

STATUS

approved

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Last modified March 27 02:45 EDT 2017. Contains 284144 sequences.