Sequence Problem

[Guest]

Hi,

Would anyone be able to solve the below sequence, as an equation where V, X and Y are the inputs and Z is the output?

V X Y Z
4 0 1 1
1 0 2 2
0 1 2 2
1 1 2 1
2 0 2 2
0 2 2 2
2 1 2 2
1 2 2 2
2 2 2 2
3 0 2 2
0 3 2 2
1 3 2 2
2 3 2 2
4 0 2 2
1 0 3 3
0 1 3 3
1 1 3 3
2 0 3 6
0 2 3 6
2 1 3 6
1 2 3 7

Any help would be much appreciated!

[Guest]

If you are looking for a simple pattern and equation that describes the data I couldn't find one.

However you can always force an equation to fit the data, the downside is that the resulting equation is usually not simple. Also such an equation is very poor at predicting the outcome for inputs that aren't already on your list.

To see how we can force an equation to fit the data consider the term
[tex]V(V-1)(V-2)(V-3)(V-4)X(X-1)(X-2)(X-3)(Y-1)(Y-2)(Y-3)[/tex]
This term is [tex]0[/tex] for every input given in the list because
[tex]V[/tex] is [tex]0,1,2,3[/tex] or [tex]4[/tex],
[tex]X[/tex] is [tex]0,1,2[/tex] or [tex]3[/tex], and
[tex]Y[/tex] is [tex]1,2[/tex] or [tex]3[/tex].

Now consider the term
[tex]V(V-2)(V-3)(V-4)X(X-1)(X-2)(Y-1)(Y-3)[/tex]
By removing some of the brackets, this new term is [tex]0[/tex] on all the inputs except when [tex]V=1,X=3,Y=2[/tex].
When [tex]V=1,X=3,Y=2[/tex] the term equals [tex]36[/tex]. In the table we see [tex]Z=2[/tex], so if I multiply the above term by [tex]2/36[/tex] I get a new term
[tex](2/36)V(V-2)(V-3)(V-4)X(X-1)(X-2)(Y-1)(Y-3)[/tex]
which is [tex]0[/tex] for all inputs except [tex]V=1,X=3,Y=2[/tex] at which point it gives the right answer of [tex]2[/tex].

I can find such a term for every entry in the table. When I add up all these terms the resulting equation will match the output [tex]Z[/tex] exactly, because for every input all the terms except one will be 0 and that non-zero term has a coefficient which was chosen to match [tex]Z[/tex].

As I said before this is a very poor way to form an equation that predicts data not in the list, but if you just want something that fits the data this is one way to go.

Hope this helped,

R. Baber.