Number of prefixes that match suffixes Consider all pairs of strings $A$ and $B$ each of length $n$ where $A_i, B_i \in \{0,1\}$.   We compare all the prefixes of $A$ with all the suffixes of $B$ of the same length and report where they match exactly. For example,  consider $A=01011$ and $B=11010$.  We have $0 = 0$, $01 \ne 10$, $010 = 010$, $0101 \ne 1010$ and $01011 \ne 11010$.  
For strings $A$ and $B$ we will report this sequence of matches and non-matches as a tuple of $1$s and $0$s where $1$ indicates a match and $0$ indicates a non-match. So in the above case we get $(1,0,1,0,0)$.

Taken  over all $2^{2n}$ different pairs of $A$ and $B$ strings, what
  is the total number of different tuples of matches and
  non-match one gets?

For $n = 1,\dots, 10$ the exact numbers are $2,4,7,11,17,25,35,48,65,86$.

The sequence $1,2,4,7,11,17,...$ seems to be
$$\sum_{i=0}^n\kappa(i)$$where $\kappa(i)$ is the kappa (auto)correlation function.
No formula for $\kappa(i)$ is known. It appears in OEIS as A005434 and was introduced in this article:


*

*L. J. Guibas and A. M. Odlyzko, Periods in Strings, Journal of Combinatorial Theory A 30:1 (1980) 19-42

