How do I derive a function that would store non-zero elements of a lower triangu

Question

How do I derive a function that would store non-zero elements of a lower triangular matrix into a sequence by row-order?

Let's say I have an n x n matrix where n = 5.

I already have the max number of non-zero elements, which is determined by the formula:

How do I figure out the formula, k, or f(i,j) where i represents the row and j represents the column, that would assign matrix element a[i,j] to its appropriate position in an array by order of row?

ex. a formula that would take a21 in the matrix above and generate the number 5, as it is the fifth matrix element by row-order.

EDIT: I am looking for a non-code expression for this formula, i.e. a mathematical formula similar to f(i,j) = (i-1)+3j that would find out the array position for each one of the matrix indices.

Explanation / Answer

you can get the appropriate position to store in array by formula

(i(i+1)/2)+j
if you want to store array from 0 index

or
(i(i+1)/2)+j+1
if you want to store array from 1 index

i consider storing from 0th index.. you can use either of formulae based on your
requirement...

where i is the row indices
and j is the column indices

example
you want to store a(0,0)

by above formula we get 0 so you store this in array 0 index

now a(2,1)

=(2(2+1)/2)+1

=4 you can store this element in array 4th index...

like wise.. if you consider storing from index 1.. formula 2 gives you 5

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