Let n be an odd number. Suppose we have the sequence s:=n,n+2,n+4,⋯. The goal is to associate this sequence with two other sequences, a and b. I will show how a and b are related to the original sequence with an example:
Both a and b act as some sort of a "counter": a counts from 0 to n−1 and resets back to 0 if it equals n. b starts from 1 and increases by 2 every time a resets.
Instead of having to depend on the previous values, I wish to find these counters as a function of s and n only.
As we notice, the sequence a is very similar to modulo, with a slight modification, i.e.,
Can b be found as a function f(s,n) too? I suspect that this would involve the quotients of s but I could not come up with any.
ahas gone through, thenbis just that amount, times 2, plus 1.ais the same as working withssince we already established it as a function ofsandna. Instead ofa = s * ((1+n) / 2) % n, replace modulo with division:cycleCountOf(a) = s * ((1+n) / 2) / n