In how many ways can you split a string of length n such that every substring has length at least m? Suppose you have a string of length 7 (abcdefg) and you want to split this string in substrings of length at least 2.
The full enumation of the possibilities is the following:
ab/cd/efg
ab/cde/fg
abc/de/fg
abc/defg
abcd/efg
abcde/fg
abcdefg

And perhaps some others.
In general, if we have a string of length n and we want to split the string in substrings of length at least m, in how many ways can we achieve this?

We can model the situation with generating functions. In order to do so we consider binary strings consisting of $0s$ and $1s$. Let $$0^\star=\{\varepsilon,0,00,000,\ldots\}$$ denote all strings containing only $0$s with length $\geq0$. The empty string is denoted with $\varepsilon$. The corresponding generating function is $$x^0+x^1+x^2+\ldots=\frac{1}{1-x},$$ with the exponent of $w^n$ marking the length $n$ of a string of $0s$ and the coefficient of $x^n$ marking the number of strings of length $n$.

We encode substrings of length at least $m(\geq 2)$ as strings starting with $1$ followed by at least $m-1$ zeros. So, each substring has the form
  \begin{align*}
(1)(0^{m-1})(0^\star)
\end{align*}
  with generating function
  \begin{align*}
\frac{x^m}{1-x}
\end{align*}

$$ $$

Since each string consists of one or more substrings, the strings can be encoded as
  \begin{align*}
(10^{m-1}0^\star)(10^{m-1}0^\star)^\star
\end{align*}
  with generating function
  \begin{align*}
\frac{\frac{x^m}{1-x}}{1-\frac{x^m}{1-x}}=\frac{x^m}{1-x-x^m}\qquad\qquad m\geq 2
\end{align*}

$$ $$
OPs example with $m=2$ results in a generating function for the Fibonacci Numbers
\begin{align*}
\frac{x^2}{1-x-x^2}=x^2+x^3+2x^4+3x^5+5x^6+8x^7+\mathcal{O}(x^8)
\end{align*}
So, the coefficient of $x^7$ is equal to $8$ corresponding to eight different strings of length $7$ with substrings of length at least $2$. These strings are 
\begin{align*}
&1010100\quad\quad(2,2,3)\\
&1010010\quad\quad(2,3,2)\\
&1001010\quad\quad(3,2,2)\\
&1010000\quad\quad(2,5)\\
&1001000\quad\quad(3,4)\\
&1000100\quad\quad(4,3)\\
&1000010\quad\quad(5,2)\\
&1000000\quad\quad(7)
\end{align*}
