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Supplementing my previous question I decided to ask a new one which is more general.

So is given a natural even number $k$ and a function $f_k:\mathbb{N}_0\rightarrow\mathbb{N}_0:n\rightarrowtail(n \;\text{mod} \; k) \rightarrowtail \text{miracle} \rightarrowtail f_k(n)$ such that:

Here comes the miracle explanation (how functions acts):

  • $k = 2$

    0 -> 0 1 -> 0 
  • $k = 4$

    0 -> 0 1 -> 1 2 -> 1 3 -> 0 
  • $k = 6$

    0 -> 0 1 -> 1 2 -> 2 3 -> 2 4 -> 1 5 -> 0 
  • $k = 8$

    0 -> 0 1 -> 1 2 -> 2 3 -> 3 4 -> 3 5 -> 2 6 -> 1 7 -> 0 

If $k$ would be odd, it would be simple: $f(n) = |\frac{k}{2} - |n - \frac{k}{2}|\;|$ (fraction without decimal part). But for even $k$'s I can't figure out something, so could you please help me to express this function (must be something with modulo, absolute value, Gauss brackets of kind of this).

Thanks in advance!

Cheers

1 Answers 1

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Just use $(k-1)/2$ instead of $k/2$: $f(n)=\Bigg\lvert\frac{k-1}2-\bigg\lvert n-\frac{k-1}2\bigg\rvert\Bigg\rvert.$