TITLE: Compute index from 2d array to find correspondent index in a flat array
QUESTION [2 upvotes]: I am a programmer (not a mathematicians), and I am encountering a problem in my software which, if possibly solved mathematically would save a lot of performance issues.
I have an array of arrays (A):
[0,0]
[1,0][1,1]
[2,0]
[3,0][3,1],[3,2]
[4,0][4,1]

the notation [x,y] is exactly the kind of notation used by the system I am working on.
Besides I have another array (B) containing the same 'kind of ordered' objects but disposed at root level:
[0,0] formerly [0,0] 
[0,1] formerly [1,0]
[0,2] formerly [1,1]
[0,3] formerly [2,0]
[0,4] formerly [3,0]
[0,5] formerly [3,1]
.......

The problem I am facing is, given the A array index, how can I retrieve the index of the same element in B ? Which is, if my hypothetical function f(x,y) is accepting [2,0]
A-index = [2,0]

I would like to have in return 
B-index = [0,3]

The instruments I have at hand are a function f(x) that return objects count in A at specific index, say:
f(2)=1 object count
f(3)=3 objects count
...

I also know the total number of object, in A and B which are the same.
I am posting here also because some suggested to use an integral, but honestly, I am a bit lost, and I do not have any instruments to verify this possible suggested solution.
Please feel free to correct the question in its syntax (especially the subject), or help me to rephrase in a more scientific way, and if you think this it's not a mathematical question, you can delete it anytime.
I did tried to search in forum, but due to my lack of knowledge I didn't manage to recognize a post with same problem.

REPLY [0 votes]: Let $F$ denote the result of summing up $f$, i.e.
$$
F(n)=\sum_{k=0}^{n-1}f(k)\;.
$$
Then the B-index for A-index $[i,j]$ is $[0,F(i)+j]$.
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