# How can I find out what row and column a cell resides in when I populate a matrix diagonally?

I am populating a matrix diagonally. Check out these three examples for clarification:

1  2  4  7  11 16
3  5  8  12 17 22
6  9  13 18 23 27
10 14 19 24 28 31
15 20 25 29 32 34
21 26 30 33 35 36

1  2  4  7  11 16 22 28
3  5  8  12 17 23 29 34
6  9  13 18 24 30 35 39
10 14 19 25 31 36 40 43
15 20 26 32 37 41 44 46
21 27 33 38 42 45 47 48

1  2  4
3  5  7
6  8  10
9  11 13
12 14 16
15 17 19
18 20 21


Is it possible to create a formula, fn(columns, row, i) = (x, y), so that you can derive what column and row a specific number in the sequence resides in? So for instance, looking at the first example, we would get something like fn(6, 6, 29) = (4,5)

Any help would be greatly appreciated, thanks.

For the first section where diagonals are growing in size, number the diagonals with the variable $d$ so that $d = x + y - 1$. Now $d$ must be the smallest integer for which $1 + 2 + \ldots + d \geq i$. A bit of algebra gives you the following expression for $d$.
$$d = \left\lceil{\frac{\sqrt{8i +1} - 1}{2}}\right\rceil$$
The value of $x$ can be calculated similarly so that you can calculate $y$. The other sections can be computed similarly.