Catalan number sequence problem

Question

I have a hw problem where I am asked explain why $50 - 20\sqrt{6}$ which is equal to $1.01020514\cdots $ looks like a Catalan Sequence (decimal part) How would I even begin to explain this? My usual approach to problems like this is write out a few small cases and then generalize and explain. But something like this, I have no idea where to begin.

Answer 1

The generating function for the Catalan numbers is given by \begin{eqnarray*} c(x)=1+x+2x^2+5x^3+14x^4+ \cdots = \frac{1-\sqrt{1-4x}}{2x}. \end{eqnarray*} Substitute $x= \frac{1}{100}$ and we have \begin{eqnarray*} 1.01020514 \cdots = 50-20\sqrt{6}. \end{eqnarray*}

Edit: \begin{eqnarray*} &1&+&x&+&2x^2&+&5x^3&+&14x^4&+& \cdots &=& \frac{1-\sqrt{1-4x}}{2x} \\ =&1&+&0.01&+&0.0002&+&0.000005&+&0.00000014&+& \cdots &=& 50-20\sqrt{6}. \\ \end{eqnarray*}