Fredrick puzzled:

> Is there a graph for every possible monotonic sequence?

No, there's not one for

\[ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, \dots \]

I leave it as a puzzle to figure out why!

I don't think "monotonic" is especially relevant here. For example, it's pretty easy to find graphs where the number of paths from \\(x\\) to \\(x\\) goes like

\[ 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, \dots \]

or

\[ 1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, \dots \]