Home

Problem Statement

We have an integer sequence $A$, whose length is $N$.

Find the number of the non-empty contiguous subsequences of $A$ whose sums are $0$. Note that we are counting the ways to take out subsequences. That is, even if the contents of some two subsequences are the same, they are counted individually if they are taken from different positions.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $-10^9 \leq A_i \leq 10^9$
• All values in input are integers.

Sample Input 1

6
1 3 -4 2 2 -2

Sample Output 1

3

There are three contiguous subsequences whose sums are $0$: $(1,3,-4)$, $(-4,2,2)$ and $(2,-2)$.

Sample Input 2

7
1 -1 1 -1 1 -1 1

Sample Output 2

12

In this case, some subsequences that have the same contents but are taken from different positions are counted individually. For example, three occurrences of $(1, -1)$ are counted.

Sample Input 3

5
1 -2 3 -4 5

Sample Output 3

0

There are no contiguous subsequences whose sums are $0$.