Binary strings with no three equal consecutive digits

Blocks. Split a string into maximal runs of equal digits. The condition says each run has length $1$ or $2$. Conversely, the first digit and the sequence of run lengths determine the string.

So $a_n=2c_n$, where $c_n$ is the number of compositions of $n$ into parts $1$ and $2$.

Recurrence for $c_n$. Look at the last part: it is $1$ (the rest is a composition of $n-1$) or $2$ (the rest is a composition of $n-2$). Hence $c_n=c_{n-1}+c_{n-2}$, with $c_1=1$ and $c_2=2$. With $F_1=F_2=1$, $c_n=F_{n+1}$ by induction.

Value. $a_n=2F_{n+1}$; for $n=10$, $F_{11}=89$, so $a_{10}=178$.

Check. A direct computer count gives $2,4,6,10,16,26,42,68,110,178$ for $n=1,\dots,10$.

Lesson. Encode a string by its run lengths; restrictions on runs become restrictions on compositions.