In a system of two linear equations with two unknowns, there are three "arrangements" that we can see when we graph the two lines in the $xy$ plane:
- The two lines intersect at a single point (one solution to the system).
- The two lines are parallel and never intersect (no solution to the system).
- The two equations describe the same line (infinitely many solutions to the system).
When we move up to a system of three linear equations with three unknowns, now we have three planes in space, and there are eight distinct arrangements of the three planes:
If we consider one linear equation with one unknown, I suppose it makes sense to say that there is one arrangement, which is a single point on the real number line.
So, for one, two and three unknowns, we have the start of a sequence: $1, 3, 8, ...$
I am interested in how this sequence continues. I have searched in vain on the OEIS. Alas, there are many sequences that have $1, 3, 8,...$ and I'm not sure which, if any, are the right one.
This one: https://oeis.org/A001792 looks like it might be it, because the comments say that sequence is related to matrices in certain ways. Also I expect a formula for this sequence should involve powers of 2. But, I wouldn't bet any money on it.
Is there a sequence with a simple formula to work out the number of arrangements?