There are a few good reasons why the dependence in the solution is on ka, κa and {κ}_{0}a: These are all dimensionless numbers, and mathematical relations can never depend on parameters that have a dimension! For the case of the even solutions, the ones with {B}_{2} = 0, we find that the number of bound states is determined by how many times we can fit 2π into {κ}_{0}a. Since {κ}_{0} is proportional to (the square root) of {V }_{0}, we find that increasing {V }_{0} increases the number bound states, and the same happens when we increase the width a. Rewriting {κ}_{0}a slightly we find that the governing parameter is \sqrt{{2m\over {ℏ}^{2}} {V }_{0}{a}^{2}}, so that a factor of two change in a is the same as a factor four change in {V }_{0}.
If we put the two sets of solutions on top of one another we see that after every even solution we get an odd solution, and vice versa.
There is always at least one solution (the lowest even one), but the first odd solution only occurs when {κ}_{0}a = π