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