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

Anthony W
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    Welcome to Mathematics SE. Take a tour: math.stackexchange.com/tour. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context by stating what you understand about the problem, what you've tried so far, etc.; both to show you are part of the learning experience and to help us guide you to the appropriate help. – Spandan Kukade Aug 23 '21 at 19:20
  • There are infinitely many ways. Take for example $N=10$. Use the digits of $\pi$ to create a non-periodic sequence – Andrei Aug 23 '21 at 19:21

1 Answers1

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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 $g(n)=f(|n|)$.

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.

RavenclawPrefect
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