Integer sequences which quickly become unimaginably large, then shrink down to "normal" size again?

Question

There are a number of integer sequences which are known to have a few "ordinary" size values, and then to suddenly grow at unbelievably fast rates. The TREE sequence is one of these sequences, which starts "1, 3" and then grows to an unimaginably large value which completely dwarfs even things like Graham's Number. Another example is given by the sequence of Ackermann numbers, which also has an extremely large third term, though not as large as that of TREE(3).

I'm interested in a variant of the above concept: integer sequences which seem to start off normally, then have one or a few values which then become mind-bogglingly large, and then which end up going back to "ordinary-sized" values for the rest of the sequence. Does anyone know of things like this which arise "naturally," perhaps in the context of graph theory or combinatorics or something similar?

Obviously one can construct sequences that fit this pattern by splicing things together, but I'm mostly interested in the case where this behavior somehow occurs in some kind of natural integer sequence.

Answer 1

Goodstein sequences provide a fine example.

Answer 2

The continued fraction expansion of $\pi$ begins: $3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+\frac{1}{1+\frac{1}{1+\ldots}}}}}}$.

In condensed form and with more digits: $\pi = [3: 7, 15, 1, 292, 1, 1, 1, 2, \ldots]$. The $292$ is the interesting element of the sequence, and it corresponds to the following good rational approximation of $\pi \approx \frac{355}{113}$, comes from truncating the continued fraction at $15+1$, just before the $292$. Thanks to Gerry Myerson to pointing out my error here.

Does this sequence answer your question? Maybe. There is a theorem from real analysis using the ideas of ergodic theory that states, for almost every real number $x \in \mathbb R$, the natural number $n$ appears in the continued fraction expansion of $x$ with frequency $log_{2} \left ( \frac{(n+1)^{2}}{n(n+2)} \right )$.

This quantity clearly goes to zero as $n$ grows. However, I may be wrong but I don't think this forces the sequence to go to zero, but it certainly doesn't go to infinity.

The prove this fact, consider the Gauss measure $\nu (E) = \frac{1}{log(2)} \int_{0}^{1} \frac{1}{1+t} dt$, on $([0,1]\setminus \mathbb Q)$. Also consider the transformation $U(x)= \{\frac{1}{x}\}$, where $\{\}$ denotes fractional part. This transformation is ergodic, and from here proving the theorem is a straightforward application of the ergodic theorem.