The important property of the golomb sequence is that \texttt{golomb}_i is the number of occurances of i in the golomb sequence. I generate the golomb sequence up to the point where what I have generated is enough to describe the whole sequence. As a result of the property we know that
\texttt{golomb}_i=k\iff\sum_{j=1}^{k-1}\texttt{golomb}_j<i\text{ and } \sum_{j=1}^k\texttt{golomb}_j\geq i
(I may have an off by one error in the above formula, but it’s the idea it represents that matters)
So for indexes up to 10^{10} we can use the cumulative golomb sequence to find the value of the golomb sequence there. And it turns out that the cumulative golomb sequence goes past 10^{10} somewhere before 2\cdot 10^6
By the way, if you want to know why half of your reply turned to italic, read this