Growing multiples of variable combinations I need to find growing combinations of variables a, b, c etc. I don't know names of math things in English well, so my description of the problem will be in pseudocode. This is for a programming problem, but I want to understand the underlying mathematics. I would highly appreciate proper names of techniques used so I can look them up further.

Lets say we have a list of variables

L (a, b, c)


And one single variable l which is the maximum desired count of a single variable from L.

I need to find every combination of elements from L multiplied by numbers from 0 to l.

So, for l = 3 I expect the following result:

1a0b0c
0a1b0c
0a0b1c
1a1b0c
1a0b1c


... and so on until ...

3a3b3c
You can think of these combinations as 3-digit numbers in base $l+1$. For example, if $l=9$ then the solutions to your problem are in one-to-one correspondence with 3-digit decimal numbers, from 000 to 999. There are in total $(l+1)^3$ of them.

As an aside, it is not clear why you omitted 0a0b0c.