Write the first $6$ terms of the sequence that is described by each of the recurrence relations below:

$($a$)f_1=1,f_2=3,$ and $f_n=5f_{n-1}-2f_{n-2}$ for $n3.$

$($b$)g_1=4$ and $g_2=5.$ The rest of the terms are given by the formula $g_n=n{g}_{n-1}+g_{n-2}.$