[proof=Growth rate for $n = 5$ (finite)] For $n = 5$, there are finitely many restricted cluster variables. Code used to find the restricted cluster variables (requires this version of sage):

[/proof]

[proof=Growth rate for $n = 6$ (exponential?)] Computed up to depth 15. Code used to find the restricted cluster variables (requires this version of sage):

Scatter plot of data:

[/proof]

[proof=Growth rate for $n = 7$ (finite)] For $n = 7$, there are finitely many restricted cluster variables. [/proof]

[proof=Growth rate for $n = 8$ (finite)] For $n = 8$, there are finitely many restricted cluster variables. [/proof]

[proof=Growth rate for $n = 9$ (cubic?)] For $n = 9$, we seem to have cubic growth rate.

[/proof]

[proof=Growth rate for $n = 10$ (finite)] For $n = 10$, there are finitely many restricted cluster variables. [/proof]