You can see how the difference between two consecutive terms is constantly increasing:
[tex]a_1 = 4[/tex]
[tex]a_2 = 7 = 4+3[/tex]
[tex]a_3 = 12 = 7+5[/tex]
[tex]a_4 = 19 = 12+7[/tex]
So, for the next terms we'll have to add +9, +11, +13 and so on.
Also, note that [tex]a_2[/tex] is obtained by adding the 2nd odd number to [tex]a_1[/tex], [tex]a_3[/tex] is obtained by adding the 3rd odd number to [tex]a_2[/tex], and so on.
So, the recursive formula is
[tex]a_n = a_{n-1}+(2n-1),\quad a_1=1[/tex]
For the explicit formula, recall that the sum of the first n odd numbers is n squared. Taking into account the fact that we're not starting from 1, we have
[tex]a_n = n^2+3[/tex]
