Hmm, this problem seems familiar

For $1 \leq n \leq 6$ , $X_n = \sum_{k=1}^{n} N_{n,k}/6^k$ where $N_{n,k}$ is the number of writing k numbers (from 1 to 6) so that they add up to $n$ . Since $1 \leq n \leq 6$ , this is just the number of k-tuples of positive integers $(a_1, …, a_k)$ such that $a_1 + … + a_k = n$ . So $N_{n,k} = (n-1 \text{ choose } k-1)$ (using the usual stars and bars method).

For $n>6$ , note that $X_n = \sum_{k=1}^{6} X_{n-k} / 6$ . We have determined $X_n$ and it is straightforward to show that the maximum is attained at $n=6$ with $X_6 = 16807/46656$ . For example, it might be helpful to show that $X_n = (2+r_1^n+ … + r_5^n)/7$ where $r_1,..., r_5$ are the five complex roots of $6 x^6-x^5-x^4-x^3-x^2-x-1$ which are not one, and note that $|r_i| < 1$ .