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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. 6
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
FORMULA
z0 = a+A272876*i satisfies the equations (1+z0)^i = z0, (1+z0)*z0^i = 1.
EXAMPLE
0.6738813311078755157802311904681019338764503347933725454899813516...
MATHEMATICA
RealDigits[Re[z /. FindRoot[(1 + z)^I == z, {z, 0}, WorkingPrecision -> 120]]][[1]] (* Amiram Eldar, May 26 2023 *)
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
Sequence in context: A197141 A139350 A092560 * A018248 A146485 A049254
KEYWORD
nonn,cons
AUTHOR
Stanislav Sykora, May 15 2016
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)