I'm having a bit of a hard time figuring this out, but I have a family of graded rings with the following Poincare series: $$\frac{1+t}{1-(n-1)t}$$ where $n$ is a positive integer. Assume the coefficients are in a field. Is it possible for this ring to be generated by degree $1$ elements? I know for $n=1$ and $n=2$ it is generated by degree $1$ elements. But is it true for larger $n$?
If not, can it be finitely generated at all?
I'm thinking the answer is probably no, since I saw a theorem that states that the Poincare series has to have the form $$\frac{Q(t)}{\displaystyle\prod_{i=1}^q (1-t^{d_i})}$$ and this series does not have this form if $n>2$.
For a combinatorial interpretation, the $k$th homogeneous component has a basis in bijection with the set of words of length $k$ from an alphabet of $n$ letters where no two adjacent positions have the same letter, which has $n(n-1)^{k-1}$ elements for $k\geq 1$.