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

$\text{'}$ v $\text{'},\text{'}$ww$\text{'},\text{'}$ xx $\text{'},\text{'}$ yy $\text{'}$ and $\text{'}$ zzz$\text{'}.$

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

For each integer n$>=1,$ let t$\_($n$)$ be the number of legal strings with n letters. For example, t$\_(1)=1(\text{'}$ v $\text{'}$ is the only the legal string$).$

t$\_(2)=$

t$\_(3)=$

t$\_($n$)=$at$\_($n$-1)+$bt$\_($n$-2)+$ct$\_($n$-3)$ for each integer n$>=4$

where a$=,$b$=,$ and c$=$

For each integer n$>=1,$ let p$\_($n$)$ be the number of legal strings with n letters that also read the same right to left as they do left to right $($like $\text{'}$ xxvxx$\text{'}$ for example$).$

Which of the following expressions is equal to p$\_(101)?$

p$\_(50)+$p$\_(49)$

t$\_(50)+$t$\_(48)$

t$\_(50)+$t$\_(49)$

t$\_(50)+2$t$\_(48)$

p$\_(100)+$p$\_(99)$

t$\_(50)+2$t$\_(49)$

t$\_(100)+$t$\_(99)$

p$\_(50)+2$p$\_(49)$

Please answer all parts of the question.