OFFSET
1,1
COMMENTS
The exponential mapping exp(z) has in C a denumerable set of fixed points z_k with odd k, which are the solutions of exp(z) = z. The solutions with positive and negative indices k form mutually conjugate pairs, such as z_3 and z_-3. A similar situation arises also for the related fixed points of the mapping -exp(z). My link explains why is it convenient to use odd indices for the fixed points of +exp(z) and even indices for those of -exp(z). Setting K = sign(k)*floor(|k|/2), an odd-indexed z_k is also a fixed point of the logarithmic function in its K-th branch, i.e., a solution of z = log(z)+2*Pi*K*i. Moreover, an odd-indexed z_k equals -W_L(-1), where W_L is the L-th branch of the Lambert W function, with L = -floor((k+1)/2). For any K, the mapping M_K(z) = log(z)+2*Pi*K*i has z_k as its unique attractor, convergent from any nonzero point in C (an exception occurs for K=0, for which M_0(z) has two attractors, z_1 and z_-1, as described in my linked document).
The value listed here is the real part of z_3 = a + i*A277682.
LINKS
Stanislav Sykora, Table of n, a(n) for n = 1..2000
S. Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, Stan's Library, Vol.VI, Oct 2016.
Eric Weisstein's World of Mathematics, Exponential Function.
Wikipedia, Exponential function.
EXAMPLE
2.062277729598283884978486720008045951283592306704591613100984...
MATHEMATICA
RealDigits[Re[-ProductLog[-2, -1]], 10, 105][[1]] (* Jean-François Alcover, Nov 12 2016 *)
PROG
(PARI) default(realprecision, 2050); eps=5.0*10^(default(realprecision))
M(z, K)=log(z)+2*Pi*K*I; \\ the convergent mapping (any K)
K=1; z=1+I; zlast=z;
while(1, z=M(z, K); if(abs(z-zlast)<eps, break); zlast=z);
real(z)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Stanislav Sykora, Nov 12 2016
STATUS
approved