Say we have an arithmetic progression in $Z_n$ like $3, 6, 9, 12, ...$ etc. If you move a sliding window of at least 3 values over the progression the 'density' in that subset compared to if the window 'had no holes', is always the same, $\frac{1}{3}$ (I think the term I'm looking for is Natural Density, but this is a new concept for me so I'm not sure if I'm using it 100% correctly).
Arithmetic progressions also have the nice property that they can be described with a fixed amount of space, e.g. for an arithmetic progression you only need to know the starting number and the constant difference to derive the rest of the numbers in the progression.
What types of monotonically increasing progressions can be described with fixed space are known that have "varying" density? By "fixed space" I mean to avoid a hard coded list, instead focusing on sequences that can be generated from some finite set of starting numbers. By "varying" I mean:
- A sliding window over the values should not see the same density every time, and that the density seen should increase and decrease, so the density should not be strictly monotonically increasing/decreasing, even though the progression is. So for example a sequence of triangular numbers would not work, because they only ever become further apart.
- I should be able to choose a window size vastly greater than the amount of space used to describe the progression and still have it vary (so no cycling through a fixed list of differences).
Apologies if I have not formalized this enough :) I think I'm stumbling on something that must be studied already but I don't know the terminology.
Edit: I was trying to keep things succinct but more constraints are needed. My ultimate goal is to use these sequences on a computer, so sequences that are more easily computable have preference. I'd also like to be able to easily compute the Nth element, without calculating every prior element.