$2$

To play a game, $n$ students stand in a row, and they can rearrange themselves in the following way. Every student can either stay in the same spot or move one spot to the left or right from where he$/$she originally stands. For example, for $4$ students standing in a row as ABCD, they can rearrange themselves as BADC or ACBD, but NOT ACDB. Let $L_n$ be the number of ways of such rearrangements $($including the case where everyone chooses to stay in the same spot$).$

$($a$)$ By considering a suitable recurrence relation, find $L_{16}.$

$4$

$($b$)$ If $L_k-=1(mod3),$ find the number of possible values of $k,$ for $k2024?$

$[3]$