$$a_2=a_1 +2=a_1+a_1$$ $$a_3=a_2 +2=a_2+a_1$$ $$a_4=a_3 +4=a_3+a_2$$
The pattern is $a_1=2,$ $a_n = a_{n-1}+a_{[\frac{n}{2}]}, n\geq 2$, where $ [\cdot ]$ is the floor function.
How to get the general formula of $a_n$?
I find $n=1,2,3, \ldots ,10$, then $a_n = 2,4,6,10,14,20,26,36,46,60$. But I don't know how to get the general formula.
You can find some valid formulae as $a_n = a_{n-1} + a_{[\frac{n}{2}]}$ (using your notation) by taking a look at the "FORMULA" section of the OEIS sequence A000123