Solution:

First choose which planet sits in each chair; the individuals from each planet can then be assigned to their chairs in $5!$ ways apiece, so $N$ counts the planet patterns. The adjacency rules say exactly that, reading clockwise, each maximal block of Martians must be followed by a block of Venusians and then a block of Earthlings before Martians can appear again. Since chair $1$ holds a Martian and chair $15$ holds an Earthling, the chairs from $1$ to $15$ consist of the pattern (Martian block, Venusian block, Earthling block) repeated $k$ times, for some $1 \le k \le 5.$

For a given $k,$ each planet's five members are distributed into $k$ nonempty blocks in order, and the number of ways to write $5$ as an ordered sum of $k$ positive integers is $\binom{4}{k-1}.$ The three planets' block sizes are independent, so $$N = \sum_{k=1}^{5} \binom{4}{k-1}^3 = 1^3 + 4^3 + 6^3 + 4^3 + 1^3 = 346.$$