Generating function for modeling the number of ways to select five integers from 1 to n where no two are consecutive So, I guess divide it into six sections representing the difference between the numbers, so the first is $0\rightarrow n$, the second through fifth are $1\rightarrow n$ and the sixth is $0\rightarrow n$ again, and that becomes a generating function of $((1+x+x^2+\cdots+x^n)^2)((x^2+\cdots+x^n)^4)$. Does this make sense, and what coefficient represents what value of $n$?

Let $f_{k,n}$ be the number of ways to select $k$ integers in $\{ 1, 2, ... n \}$ such that no two are consecutive.  There are two ways to do this: either the largest one is not $n$, and then you choose $k$ integers in $\{ 1, 2, ... n-1 \}$, or the largest one is $n$, and then you choose $k-1$ integers in $\{ 1, 2, ... n-2 \}$.  This gives
$$f_{k,n} = f_{k,n-1} + f_{k-1, n-2}.$$
The generating function you want is $\sum_{n \ge 0} f_{5,n} x^n$, but I think it will be easier to find the bivariate generating function $\sum_{n,k \ge 0} f_{k, n} y^k x^n$ first.  Do you know how to do this from a recursion?
