I was doing research on whether an equation can be formed about the Mobius Strip on the basis of how many times it is cut (thirds, fourths, fifths, etc.). I started with 0 cuts. This is what I got: 0 cuts, 0 two-sided loops.. 1 cut, 0 2 sided loops. 2 cuts, 1 2-sided loop, 3 cuts, 2 2-sided loops. 4 cuts, 2 2-sided loops. Essentially, the equation is; 0,0,1,2,2,3,3,4,4,5,5, etc.
It's pretty easy to see that the formula ⌈n/2⌉ "almost" gives the correct sequence. The only case it doesn't handle is n=1. So one can augment the formula a bit to take care of that case: (⌈n/2⌉)sgn(n−1)takes care of n=1. (sgn is the sign function). Because sgn(n−1) gives -1 if n=0, we need to check that case. Fortunately sgn(−1)(⌈0/2⌉)=0. But what is the formula for this equation? (⌈n/2⌉)sgn(n−1) or ⌈n/2⌉ or otherwise?
Look at the secquence $\{x\}=\{\frac{i-1}{2}\}=\{0.0,0.5,1.0,1.5,2.0,...\}$. Then rounding this down, gives the wanted sequence.