Finding the number of distinct sub-strings in a binary string. Whilst solving a question, I have come across a problem regarding the maximal number of possible distinct $k$-length binary sub-strings in an $n$-length binary string.
My thought process was that if you take some $n$-length binary string, then the number of possible sub-strings could be found as follows:
$$\sum_{k=1}^{n}{n-k+1}=n^{2}-\frac{n(n+1)}{2}+n=\frac{n^{2}+n}{2}$$
But this doesn't take into account the maximum possible number of distinct sub-strings, which will be $2^{k}$, when we're looking for sub-strings of length $k$.
Does anyone have any suggestions as to how I could find the maximal number of distinct sub-strings (of length $k\le n$) contained within some binary string of length $n$?
Thanks in advance!
 A: I took my own advice: it's "Maximal number of distinct nonempty substrings of any binary string of length n." It seems that there is a question about this sequence that is very simple to state but is still open. 
A: "On the maximum number of distinct factors of a binary string" (also here), by Jeffrey Shallit, proves that the answer (the attained maximum number of distinct factors of a length-$n$ binary string) is $$\binom{n-k+1}{2}+2^{k+1}-1,$$ where $k$ is the unique integer such that $$2^k + k - 1\ \le\ n\ \lt\ 2^{k+1}+(k+1)-1.$$ (This counts the empty string as a factor of every string.) 

NB: A "closed form" for the answer can be found by noting that $k$ is just $\lfloor x \rfloor$, where $x$ is the real number such that $$2^x + x - 1=n.$$ Now, the substitutions
$$\begin{align}
X &:= \log(2)\ (n-x+1)\\
Y &:= \log(2)\ 2^{n+1}
\end{align} 
$$
transform this equation into 
$$X\ e^X = Y$$
whose solution is
$$X = W(Y),$$
where $W$ is the (principal branch of) the Lambert $W$ function.
Therefore,
$$\log(2)\ (n-x+1) = W(\log(2)\ 2^{n+1})$$
giving
$$k = \left\lfloor n+1-\frac{W(\log(2)\ 2^{n+1})}{\log(2)}\right\rfloor.$$

NB: Some comments at the OEIS link refer to the above results as "conjectures", in spite of the proof given in the cited paper.
