Intuition Behind a Decimal Representation with Catalan Numbers From $0 = 0.5 - 0.5 = 0.5 - \sqrt{0.25}$, we can adjust the subtrahend slightly to obtain
$$0.5 - \sqrt{0.249} = 0.001\ 001\ 002\ 005\ 014\ 042\ldots$$
where the decimal representation contains the first few Catalan numbers: $1, 1, 2, 5, 14, 42 
\ldots$
We can see even more Catalan numbers (albeit spaced apart with more $0$s) by using more $9$s. 
(For example, check the decimal representation of $0.5 - \sqrt{0.24999999999}$.)
My question is not how to show this formally; it is a straightforward problem to show how one derives, e.g., the decimal representation given above. (I'll include a derivation below for anyone who doesn't want to think this through her/himself.)
Instead, my question is: why would it make sense, intuitively, for the Catalan numbers to show up in these decimal representations?

Derivation:
Recall that the generating function for the Catalan numbers $c(x)$ satisfies $c(x) = 1 + xc(x)^2$. Rearranging, we find that $c(x) = \frac{2}{1 + \sqrt{1 - 4x}}.$ Then
$$\sum_{n = 0}^{\infty}\frac{C_n}{10^{3n + 3}} = \frac{1}{1000}\sum_{n=0}^{\infty}\frac{C_n}{1000^{n}} = x \sum_{n=0}^{\infty}C_n x^n = xc(x) = \frac{2x}{1 + \sqrt{1 - 4x}},$$
where we have simplified our computations by letting $x = \frac{1}{1000}$.
Evaluating at this value of $x$ yields $0.5 - \sqrt{0.249}$.
 A: Whenever you want to invert a function you should think about Lagrange inversion. In this case you want to invert a quadratic function. Lagrange inversion happens to have an elegant proof using trees which can be found, for example, in Stanley's Enumerative Combinatorics (Vol. II). This proof interprets an implicit identity satisfied by a generating function (such as the one satisfied by the generating function for the Catalan numbers) as recursively describing a certain kind of tree; for much more on this point of view, see Bergeron, Labelle, and Leroux's Combinatorial Species and Tree-Like Structures. Specializing to this case gets you one of the flavors of trees counted by the Catalan numbers. 
(But this is not the train of thought that would have actually occurred to me because the Catalan numbers are more familiar than Lagrange inversion; I just look for $\sqrt{1 - 4x}$ everywhere, and in this problem I see $\sqrt{ \frac{1}{4} - x}$, so... )
A: As a general fact, I think, intuitive is not an absolute notion. Working enough time on a mathematical subject has the effect of developping a number of authomatisms in the reasoning about that subject. This makes facts and constructions appear more intuitive, in that, we seem to acquire knowledge on them without use of reason. This phenomenon appears very clearly when you talk with an expert on a subject you do not know... 
