Call a string of letters "legal" if it can be produced by concatenating $($running together$)$ copies of the following strings:

$\text{'}\text{'},\text{'}\text{'},\text{'}\text{'}\text{'}\text{'}$ and $\text{'}\text{'}.$

For example, the string $\text{'}\text{'}$ is legal because it can be produced by concatenating $\text{'}\text{'},\text{'}\text{'}$ and $\text{'}\text{'},$ but the string $\text{'}\text{'}$ is not legal.

For each integer $>=1,$ let $$ be the number of legal strings with $$ letters. For example, $1=1(\text{'}\text{'}$ is the only the legal string$).$