One dimensional non-periodic tiling using N different colours

Question

What equation produces a non-periodic one dimensional sequence, comprising N different colours?

Answer 1

Without more details, it's hard to say exactly what you're looking for here, but here is an equation which produces an aperiodic two-coloring of the natural numbers:

$$f(n) = \sum_{k=0}^\infty\left\lfloor \frac{n}{2^k}\right\rfloor\pmod{2}$$

This will yield the Thue-Morse sequence, which is not just aperiodic but in fact contains no subsequence which is repeated three times consecutively.

If you want to color the integers, just use $g(n)=f(|n|)$.

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Of course, if all you want is to have a sequence without any periodic structure, you can do much more boring things, like coloring the whole line red and a single cell blue (if you wanted to make this into an "equation", just use an indicator function or something with the $\text{sgn}$ function if desired). It's not clear from your question exactly what sort of thing would satisfy your criteria.