Use of the ellipsis (...) to indicate multiple established patterns - math notation I understand the meaning of a single ellipses is to "continue the established pattern" in math speak. For example,
$1,2,...,n$
means to continue the established pattern of listing the positive integers until the $n$th integer is reached. However, is there a particular way to use ellipses when multiple patterns are present? For example, suppose I would like to indicate a list of positive integers that alternates between (1) listing all positive integers and (2) listing only the even integers every time a multiple of $10$ is reached until the list ends at $100$. Would I represent this as
$1,2,...,10,12,14,...,20,21,22,...,30,32,34,...,40,...,100$
or as
$1,2,...,10,12,14,...,20,21,22,...,30,32,34,...,40,...,...,100$
or is there some other convention I'm not aware of? One question I have is whether consecutive ellipses are used to indicate the continuation of multiple established patterns (as is the case in the second example above).
Thanks!
 A: Good question. I would use your first solution and I would not repeat ellipses at the end. That being said, the difficulty is to avoid any misunderstanding. One possible way to clarify would be to write
$$
1,2,...,10,\underbrace{12,14,...,20}_\text{even numbers},21,22,...,30,\underbrace{32,34,...,40}_\text{even numbers},41,...,100
$$
the first time you introduce your sequence. If you have several patterns, you may use underbraces indexed by "pattern 1", "pattern 2", etc. For more complicated examples, you may use
$$
u_1, v_1, u_2, v_2, \ldots
$$
where you define separately the sequences $u_n$ and $v_n$.
A: Not everything has an easy notation.  Arguably, your pattern repeats every 15 numbers ($1,2,\dots,16,18,20,\langle\text{repeat}\rangle$), but that's unwieldy to write.  On the other hand,

a list of positive integers that alternates between (1) listing all positive integers and (2) listing only the even integers every time a multiple of 10 is reached

is perfectly good mathematical English.  If you need to introduce such a list amidst a equation display (c.f. def'n 5), then there are multiple ways to do it.  One is to introduce a named sequence:

Let $\{v_n\}_{n=1}^N$ be the following sequence: list the numbers from $1$ to $100$ in order, but, each time one reaches a multiple of $10$, alternate between dropping the even numbers and not dropping them.  For example, we begin $$1,2,\dots,10,12,14\dots,20,21,22,\dots$$  (Choose $N$ so that our sequence ends with $100$.)
Then $$\langle\text{complex equation involving }\{v_n\}_n\rangle$$

Another is to use a single dots, and just include an explanatory note afterwards:

\begin{align*}
A&=\langle\text{first line}\rangle \\
&=1+\dots+100 \tag{*}\\
&=\langle\text{third line}\rangle
\end{align*} where in (*), the summands are produced by listing the numbers from $1$ to $100$, and, each time one encounters a multiple of ten $10$, alternate between dropping and not dropping the even numbers.

Your description is a little bit awkward, because it describes a way to (eventually) construct the sequence, but not to easily check whether a number appears in the sequence or not.  Here's an alternate definition, using essentially the same idea as Zubin Mukerjee's:

Let $$S=\{x:(\forall n\in\mathbb{Z})(10n<x\leq10(n+1)\Rightarrow(2\nmid n\Rightarrow2\nmid x))\}\subseteq\{1,2,\dots,100\}$$  Then the sequence in question is $S$, listed in increasing order.

