You could do it this way:
The number of length one strings (L1) = 2.
The number of length two strings (L2) = 2.
These can be trivially found out.
Now for L3, we fix the first and last positions as a_a or b_b. Now, in the blank there has to be a string of length 1, for which we have two options. Therefore, the total number of strings are (a_a)*2 + (b_b)*2 = 4.
We can extend this logic for all the further substrings.
L4 = a__a + b__b where the blank will contain strings of length 2.
L5 = a___a + b___b where the blank will contain strings of length 3.
L6 = a____a + b____b where the blank will contain strings of length 4 and so on.
When you add all these, you should get 60.