this is quite a vague question, more of a puzzle than a question. I've spotted that two problems concerning word combinatorics have the same answer. I feel like there should be a connection, but I have none.
Length of 'unwrapped' de Bruijn sequence is (in binary case) $2^{n} + n - 1$ (de Bruijn word is such a shortest word containing all words of length $n$ as subwords, e.g. for $n=2$ the word 00110)
Imagine an experiment consisting of sequence of coin tosses. We can (and we want to!) stop this truly tedious activity if we spot a predetermined sequence. It's possible to compute an expected time of this, more precisely expected time of the first occurrence of the given sequence. For example for three heads in a row, HHH, it is 14 coin tosses. Let's look at this experiment as the organizers. We have lots of subjects and for each of them we generate randomly and uniformly some $n$-word ($n$ is fixed), we are interested in the average time it takes for subjects to complete their task. More precisely, now even the predetermined sequence is chosen randomly. The answer is again $2^n + n - 1$, as is computed in here.
It works for larger alphabets as well (and more sided coins??). Is there a connection? If not precise, at least some heuristics. Or is it just a pure coincidence? (Coincidences in maths? There's no such thing I want to believe.)