Find a compact form for generating function of the sequence 4, 4, 4, 4, 1, 0, 1, 0, 1, 0, 1, 0,

The generating function for this sequence is

assuming the sequence itself is {4, 4, 4, 4, 1, 0, 1, 0, ...} and the 1-0 pattern repeats forever (as opposes to, say four 4s appearing after every four 1-0 pairs). We can make this simpler by "displacing" the odd-degree terms and considering instead the generating function,

where the coefficients of follow a much more obvious pattern of alternating 1s and 0s. Let

where is recursively given by

and explicitly by

so that

and so

8121 because

130,000 divided by 16 is 8121