Mod

The usage of Mod is straightforward, but by "offset" the Doc Center means shifting the cycle of remainders over by the offset along the number line. Normally Mod[m,n] is equivalent to the typical infix mathematical notation m mod n, i.e. it gives the remainder of m/x  (where x is a positive integer) as cycling from 0 through n-1. Offset rotates the cycle on the number line by its value. Here is the usage with default offset of 0:

In[149]:= Mod[Range@7, 3]


Out[149]= {1, 2, 0, 1, 2, 0, 1}

Now using an offset of 1, instead of cycling through {0,1,2}, the remainder cycles through {1,2,3}:

In[151]:= Mod[Range@7, 3, 1]


Out[151]= {1, 2, 3, 1, 2, 3, 1}

Using an offset of 2, instead of cycling between 0 and 2, the remainder cycles through {2,3,4}:

In[152]:= Mod[Range@7, 3, 2]


Out[152]= {4, 2, 3, 4, 2, 3, 4}

So what should happen if we use an offset of -1? The remainder should cycle through {-1,0,1}.

In[153]:= Mod[Range@7, 3, -1]


Out[153]= {1, -1, 0, 1, -1, 0, 1}