I'm not 100% confident in my explanation below, but maybe it's something like this, if we don't have prior knowledge of the asymptotic behavior of random walks:

The measured values are empirically exponential with a factor of 1.424 per indentation. If we assume that as an upper bound (which I don't think is quite right), then we have O(1.424^N) where N is the number of indentations.

Any exponent can be written in terms of any other base by multiplying by a constant. Since we often use base 2, we can choose to rewrite it in base 2 as O(2^(K*N)), where K is approximately 0.51.

Constant terms can be ignored in big O notation, so we can just drop the K to get O(2^N) (I am less sure about dropping a constant within the exponent).