Given an arbitrary sequence of nonnegative integers $x_0, \ldots, x_{n-1}$, define its difference sequence as x'_i = | x_i - x_{i+1} |, with indices mod $n$ (so $x_n = x_0$). Repeated application of this process leads to interesting behavior. For example, for $n=2^k$, it appears the process always ends with the zero sequence:
181,530,245,548,294,228,364,958 349,285,303,254,66,136,594,777 64,18,49,188,70,458,183,428 46,31,139,118,388,275,245,364 15,108,21,270,113,30,119,318 93,87,249,157,83,89,199,303 6,162,92,74,6,110,104,210 156,70,18,68,104,6,106,204 86,52,50,36,98,100,98,48 34,2,14,62,2,2,50,38 32,12,48,60,0,48,12,4 20,36,12,60,48,36,8,28 16,24,48,12,12,28,20,8 8,24,36,0,16,8,12,8 16,12,36,16,8,4,4,0 4,24,20,8,4,0,4,16 20,4,12,4,4,4,12,12 16,8,8,0,0,8,0,8 8,0,8,0,8,8,8,8 8,8,8,8,0,0,0,0 0,0,0,8,0,0,0,8 0,0,8,8,0,0,8,8 0,8,0,8,0,8,0,8 8,8,8,8,8,8,8,8 0,0,0,0,0,0,0,0
Is this a theorem? For $n$ not a power of 2, it falls into cycles where all elements are 0 or $m$ for some integer $m$ (usually $m=1$).
Have these difference sequences been studied? They seem to have a rich structure.