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 $0.5 - \sqrt{0.249}$.