Consider a binary string with $s$ ones and $m$ zeros in total.

Let's put an additional (dummy) fixed zero at the start and at the end of the string.
We individuate as a *run* the consecutive $1$'s between two zeros, thereby including runs of null length. With this scheme we have a fixed number of $m+1$ runs.

The number of different strings with the above numbers of zeros and ones is obviously
$$
\left( \matrix{ m + s \cr s \cr} \right) = \left( \matrix{ m + 1 + s - 1 \cr s \cr} \right)
$$
which corresponds to the weak compositions
of $s$ into $m+1$ parts.

The number of compositions of $s$ into $k$ non-null parts (*strong* compositions) is instead
$$ \binom{s-1}{k-1} $$
and
$$
\eqalign{
& \left( \matrix{ m + s \cr s \cr} \right)
= \sum\limits_{\left( {1\, \le } \right)\,k\,\left( { \le \,\min \left( {m + 1,s} \right)} \right)}
{\left( \matrix{ m + 1 \cr k \cr} \right)\left( \matrix{ s - 1 \cr k - 1 \cr} \right)} = \cr
& = \sum\limits_{\left( {1\, \le } \right)\,k\,\left( { \le \,\min \left( {m + 1,s} \right)} \right)}
{\left( \matrix{ m + 1 \cr m + 1 - k \cr} \right)\left( \matrix{ s - 1 \cr k - 1 \cr} \right)} \cr}
$$

So we can concentrate on strong compositions with no equal consecutive parts.

Consider the strong composition of $s$ into $k$ parts, the last of which is $r$
$$
\left[ {r_{\,1} ,\,r_{\,2} ,\; \cdots ,\,r_{\,k - 1} ,r\;} \right]
\quad \left| {\;r_{\,1} + \,r_{\,2} + \; \cdots + \,r_{\,k - 1} + r = s} \right.
$$
whose number is
$$
C_T(s,k,r) = \left[ {k = 1} \right] + \left( \matrix{ s - r - 1 \cr k - 2 \cr} \right)
= \left( \matrix{ s - r - 1 \cr s - r - k + 1 \cr} \right)
\quad \left| \matrix{ \;1 \le k \le s \hfill \cr \;1 \le r \le s \hfill \cr} \right.
$$
where $[P]$ denotes the *Iverson bracket*.

Then the sum over $r$ will correctly give
$$
\eqalign{
& \sum\limits_{r = 1}^s {C_T (s,k,r)}
= \sum\limits_{r = 1}^s {\left( \matrix{ s - r - 1 \cr s - r - k + 1 \cr} \right)}
= \sum\limits_{j = 0}^{s - 1}
{\left( \matrix{ j - 1 \cr j - k + 1 \cr} \right)} = \cr
& = \sum\limits_{\left( {k - 1\, \le } \right)\,j\,\left( { \le \,s - 1} \right)}
{\left( \matrix{ s - 1 - j \cr s - 1 - j \cr} \right)\left( \matrix{ j - 1 \cr j - k + 1 \cr} \right)}
= \left( \matrix{ s - 1 \cr s - k \cr} \right) \cr}
$$

Let's indicate with $C_G (s,p,r), \; C_B (s,p,r)$ the number of good and bad strong compositions of $s$ into $p$ parts the last of which
equal to $r$.

Then we have the relations
$$
\left\{ \matrix{
C_T (s,p,r) = \left[ {1 \le p \le s} \right]\left[ {1 \le r \le s} \right]\left( \matrix{
s - r - 1 \cr
s - r - p + 1 \cr} \right) \hfill \cr
C_G (s,1,r) = C_T (s,1,r) = \left[ {r = s} \right]\quad C_B (s,1,r) = 0 \hfill \cr
C_G (s,p,r) + C_B (s,p,r) = C_T (s,p,r) \hfill \cr
C_B (s,p,r) = \sum\limits_{k = 1}^{s - r} {C_B (s - r,p - 1,k)} + C_G (s - r,p - 1,r) = \hfill \cr
= \sum\limits_{k = 1}^{s - r} {C_B (s - r,p - 1,k)} - C_B (s - r,p - 1,r) + C_T (s - r,p - 1,r) \hfill \cr
C_G (s,p,r) = \sum\limits_{k = 1}^{s - r} {C_G (s - r,p - 1,k)} - C_G (s - r,p - 1,r) \hfill \cr} \right.
$$

In particular for the good strong compositions we can write the recurrence
$$
C_G (s,p,r) = \sum\limits_{k = 1}^{s - r} {C_G (s - r,p - 1,k)} - C_G (s - r,p - 1,r) + \left[ {1 = p} \right]\left[ {r = s} \right]
$$

After computing $C_G$, we can sum on $r$ and then go back along the previous steps
to compute the good weak compositions in terms of $s,m$ and finally the number in $n$,
i.e.:
$$
N_G (n) = \sum\limits_{s = 1}^n {\sum\limits_{p = 1}^{n - s + 1}
{\left( \matrix{ n - s + 1 \cr p \cr} \right)
\sum\limits_{r = 1}^s {C_G (s,p,r)} } }
$$
which in fact for $0 \le n \le 16$ gives
$$
0, \, 1, \, 3, \, 6, \, 12, \, 23, \, 44, \, 82, \, 153,
\, 284, \, 527, \, 978, \, 1814, \, 3363, \, 6234, \, 11554, \, 21413
$$
not counting as good the all zeros string.