How do you find the nth to the quadratic sequence of 3,8,15,24,35

Having the first term of the sequence

a (0) = 3

a (1) = 3+5=8

We realized that

a (1) = a (0) + 2⋅2+1

We also have:

a (2) = a (1) + 2⋅3+1=8+7=15

a (3) = a (2) + 2⋅4+1=15+9=24

From above we can realize that each term is the sum of the previous

term and 2 * (sequence coefficient added to 1) and 1

So the nth term will be:

a (n) = a (n-1) + 2⋅ (n+1) + 1