Let $A$ be a sequence of letters $\langle a,b,c,d,e,f \rangle$. I want to create two subsequences, one with the values with odd index and other with the values with even index: $A_\mathrm{odd} = \langle a,c,e \rangle$ and $A_\mathrm{even} = \langle b,d,f \rangle$.
My question is: is there any usual symbol that could substitute the words "odd" and "even" in the name of the subsequence?
Thanks!
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$\begingroup$"odd" and "even" are fine. Maybe in roman not italic, though: $$ A_{\rm{odd}} \ne A_{\rm{even}} $$ since the first subscript is not a product $odd$ of three factors.
$\endgroup$ 1 $\begingroup$The best I can come up with is $A_{2k+1}$ and $A_{2k}$.
$\endgroup$ 8 $\begingroup$How about $A_\mathcal O$ and $A_\mathcal E$?
To produce these: A_\mathcal O and A_\mathcal E
To say that $x$ is even is to say that $x\equiv 0 \bmod 2$ and to say that it is odd is to say that $x\equiv 1 \bmod 2$. So you could use $A_0$ and $A_1$.
But, to be honest, I prefer $A_{\mathrm{e}}$ and $A_{\mathrm{o}}$, since they're as easy to remember as $A_{\mathrm{even}}$ and $A_{\mathrm{odd}}$ but more concise.
$\endgroup$ 4 $\begingroup$I'm missing here this option:
$A_{2|}$ for even sequence, and $A_{2 \nmid}$ for odd.
As $2|a$ states that $a$ is divisible by 2, thus even. While $2\nmid a$ states that $a$ is not, thus odd.
I realise this is an old question, but I had the same question and found there was one good option missing from this list of answers so I thought I'd add it in case it could help anyone else that stumbled on this thread.
By convention, the symbols $\mathbb{Z}$ or $\mathbf{Z}$ are used to denote the set of all integers, and the symbols $\mathbb{N}$ or $\mathbf{N}$ are used to denote the set of all natural numbers (non-negative integers). It is therefore intuitive that something like $2\mathbb{Z}$ would mean all even numbers (the set of all integers multiplied by 2 becomes the set of all even numbers), and $2\mathbb{Z}+1$ would likewise mean the set of all odd numbers. If you didn't need negative numbers, then you could instead write $2\mathbb{N}$ and $2\mathbb{N}+1$, respectively.
In the context of your problem it would make sense to use this terminology to denote your sequences as either $A_{2\mathbb{Z}}$ and $A_{2\mathbb{Z}+1}$, or $A_{2\mathbb{N}}$ and $A_{2\mathbb{N}+1}$. Since your sequences consist of letters of the alphabet, then negative numbers don't make a lot of sense, so I would lean more towards using $A_{2\mathbb{N}}$ and $A_{2\mathbb{N}+1}$ as synonymous for $A_\mathrm{even}$ and $A_\mathrm{odd}$, respectively.
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