Sets are collections of mathematical objects without importance to the order or repetition. That is $$\{0,0,0,0,0,0,1,1,1,1,2,2,2,2,1,1,1,1,0\}=\{0,1,2\}=\{2,2,1,0\}=\ldots$$
If you are interested in the order then you wish to talk about sequences rather than sets. Sequences are often denoted by $\langle a_i\mid i\in I\rangle$ where $I$ is an index set which carries (usually) some natural order, at least in the case of sequences. For example $I$ can be taken as the natural numbers or a finite subset of them. If the index set is very small we can just write the sequences as $\langle a_1,\ldots,a_n\rangle$.
So we have $\langle 21,34,42\rangle$. We can treat this as a function from $\{1,2,3\}$ into some other set, that is $h(1)=21, h(2)=34, h(3)=42$. Then we can write $h(1)$ or $h_1$ for the first element of the sequence.
Best Answer
I don't recall seeing too many places that gave a specific notation to the set of even or odd numbers. Your notation of $2\mathbb N+1$ seems quite reasonable.
As with all notational problems, my usual tip is to find something that seems reasonable and simply declare it in the first few lines (or when you need to use it):