Given $a, b$, how do we find the number of distinct sums we can get from the range $[a,b]$?

Question

This is a problem from a previous math competition. $a, b$ can be anywhere in the range $[1,10^9]$ ($a \le b$).

Edit: We can choose any subset of numbers from $a$ to $b$ to form a single sum. Each number cannot be used twice for the same sum.

For example, if $a = 2, b = 5$, then one possible sum is $2 + 4 + 5 = 11$.

The possible sums we can get range from $a$ up to $\frac{b(b+1)}{2} - \frac{a(a-1)}{2}$. I got this second number by using the formula $\sum_{i = a}^{b}{i} = \sum_{i = 1}^{b}{i} - \sum_{i = 1}^{a-1}{i} $. But still, this is a really large range to be brute-forced. For example, if $a = 1, b = 10^9$, then the range to be brute-forced would contain around $10^{18}$ numbers. How do I solve this? Any help would be much appreciated :)

What are "sums from the range [a, b]", precisely? How many summands are allowed? Can there be any repetitions? — lisyarus, Aug 31 '20 at 12:45
@lisyarus I edited the question. Basically you can use any number of summands but you cannot use a number more than once. (Just like picking a subset of the numbers in $[a,b])$ — skar90, Aug 31 '20 at 13:05
What did the original problem state? Are you trying to generalize for arbitrary $a$ and $b$? — Peanut, Aug 31 '20 at 13:17
@Peanut The problem statement is the same as the one I've posted here. — skar90, Aug 31 '20 at 13:20
When you talk about "choose any subset", do you allow the subset to contain one element? For example, $5=5$ is a valid sum? And by "numbers", you mean "integers" I assume. — Vepir, Aug 31 '20 at 15:45
This is a possible duplicate of: Is it possible to easily count all the possible distinct sums from distinct numbers? — Vepir, Aug 31 '20 at 15:49

Given $a, b$, how do we find the number of distinct sums we can get from the range $[a,b]$?

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