Two variable discrete linear combination

[augusto.altman]

Two variable discrete function

Hello everybody, I come here with one question that have kidnapped my mind in the last few weeks: Can a two variable discrete function be transformed into a one variable discrete strictly increasing monotonic sequence?

Consider, for example, the discrete function: F(m,n)=Amn+Bm+Cn+D
where A,B,C,D are non-zero positive integers and constants, and m,n are positive integers (including zero) and variables.

Can that be transformed in something like G(i) where G is the ordered sequence (strictly increasing monotonic sequence of i) of the numbers resulted from the different combinations of m and n evaluated on F?

Thank you very much.
Augusto.-

 

[HallsofIvy]

Hello everybody, I come here with one question that have kidnapped my mind in the last few weeks: Can a two variable discrete linear combination be transformed into a one variable discrete strictly increasing monotonic sequence?

Consider, for example, the discrete linear combination: F(m,n)=Amn+Bm+Cn+D

This is not a linear combination.

where A,B,C,D are non-zero positive integers and constants, and m,n are positive integers (including zero) and variables.

Can that be transformed in something like G(i) where G is the ordered sequence (strictly increasing monotonic sequence of i) of the numbers resulted from the different combinations of m and n evaluated on F?

Thank you very much.
Augusto.-

Given any double indexed array of numbers, you can always create a single indexed sequence by "zig-zaging" through the array: G(1)= F(1,1), G(2)= F(1, 2), G(3)= F(2. 1), G(4)= F(3, 1), G(5)= F(2. 2), G(6)= F(1, 3), etc.

But that will not give a "strictly increasing" sequence. It sounds like what you want to do is just form the set of all F(m, n) values and order that set without regard for m and n. But even that will not give a strictly increasing sequence since your method of creating F(m, n) may have many duplicate values depending on the values of A, B, C, and D.

 

[augusto.altman]

It sounds like what you want to do is just form the set of all F(m, n) values and order that set without regard for m and n

Yes, it's a valid way of seeing it. What I want is a new function G that able to give me all the values that can be possible generated with F (from evaluating all the different combinations on n and m) but sorted increasingly.

So for example suppose that I have:
F(m,n)=100mn+110m+30n+33

Then, taking some values on m and n I obtain: (FYI: I'm aware that I will get repeated values on F on different combinations on m,n, but the following table don't reflect that)

m	n	F
0	0	33
0	1	63
0	2	93
0	3	123
0	4	153
0	5	183
1	0	143
0	6	213
1	1	273
2	0	253
AND	SO ON	INFINITELY

Then I want to get a function G(i) that gives me the following values:

i	G
0	33
1	63
2	93
3	123
4	143
5	153
6	183
7	213
8	253
9	273
AND SO ON	INFINITELY

So, seeing I want the set of values generated by F? yes, I want the values order without regard of M and N? yes. Another way of seeing it is to find a relation that enables me determine the values of m and n based on the value of i. So, back to the general case:

If m(i) = XXX, n(i) = YYY, therefore as m and n depends on i, F depends on i too (we obtain G):

F(m,n) = Amn+Bm+Cn+D = Am(i)n(i)+Bm(i)+Cn(i)+D = G(i)

Maybe there is a well known way of achieving this, I wanted to find out that. If you know a way of doing this without "sorting" the values let me know too since I may be able to do some stuff with that too.

Thanks in advance,
Augusto.-

PS: I have updated the post in order to use the proper terminology (you are right it's not a linear combination). Thank you for pointing that out.