A series such that $\sum {a_n}$ converges, but $\sum {a_{3n}}$ diverges.

It is entirely fine to define a sequence term by term, and your examples are fine. In fact $\LaTeX$ even supports this with the following environment:

a_n=
\begin{cases}
 [value 1] & [condition 1] \\
 [value 2] & [condition 2] \\
 ...
\end{cases}

For example (right click to show underlying code): $$a_n= \begin{cases} 2&\text{ if }\ 3\text{ divides } n\\ -1&\text{ otherwise} \end{cases} \qquad\text{ and }\qquad b_n= \begin{cases} 0&\text{ if }\ 3\text{ divides } n\\ 1&\text{ otherwise} \end{cases}.$$