While reading Steven Finch's wonderful book Mathematical Constants, I encountered the Trott constant which was presented as the real number such that the digits of its decimal expansion are the digits of its simple continued fraction expansion. This number was given as:
$$E=0.1084101512231113...=0+\cfrac1{1+\cfrac1{0+\cfrac1{8+\cfrac1{4+\cfrac1{1+\cfrac1{0+\cfrac1{1+\ddots}}}}}}}$$
My question is how such a number is found. After I read this I spent some time trying to show the existence of such a number by hand, but failed. I was able to establish some basic bounds but unable to take it much further. I am still in the dark about how such a number is computed. This paper from 2006 discusses the problem and mentions that the author (Trott) had been unable to establish existence or uniqueness of such numbers, the approach having been purely experimental (using Mathematica); the paper contains the code used to find $E$ but I cannot read Mathematica and am not sure of the method used. The OEIS doesn't help me.
What I was wondering was whether anyone could quickly elucidate the method used to find such a number, and also whether anyone knows of any existence or uniqueness results that have been established yet for such numbers? Finally, is anything known about the existence of such numbers in other bases (from my playing around with the problem I began to suspect that some bases might not allow such a number)?
