How many unique numbers can be obtained by adding two numbers from two different sequences?

Question

Let the two integer sequences $\{a_m\}$ and $\{b_m\}$, be defined as: $a_n+D_n=a_{n+1}$ and $b_n=a_n-k$, where $D_n$ may be any natural number (and $D_i$ may or may not be equal to $D_j$), $k$ is an integer constant, and $a_1$ is also an integer. Find how many different numbers can be obtained by adding any $a_j$ to any $b_i$.
My attempt
I found how many possible numbers I could get from the two sequences by arranging all elements of $\{a_m\}$ and $\{b_m\}$ in an addition table. The table has $m^2$ entries. But because addition of integers is commutative, it would also be diagonally symmetric, making the new amount of possible different numbers equal to $\frac {m^2+m}{2}$.
However, I noticed that by making the values of some $D_i$'s equal to each other or equal to $k$ some numbers in the table would be repeated. My best guess is that the number of repetitions in the values of $D_i$ is somehow related to the repetitions in the table, but I have no idea how to prove it.
Question
How can I find the number of repeated numbers in the table from the information above?

Answer 1

Without some information about the $D$'s there is no answer. If the $D$'s are larger than $mk$ and distinct enough, all the $m^2$ entries in the table will be distinct. The table need not be symmetric because the increments in the two axes differ. If the $D$'s all equal $k$ there will only be $2m-1$ distinct entries.