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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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    $\begingroup$ $o$ and $e$ indexes. $\endgroup$ – Yves Daoust Nov 25 '14 at 16:19
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"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.

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    $\begingroup$ Ah, the identic substitutions for „odd“ and „even”. :-) $\endgroup$ – mvw Nov 26 '14 at 9:27
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The best I can come up with is $A_{2k+1}$ and $A_{2k}$.

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    $\begingroup$ What's $k$? I don't see any $k$ in the specification of the problem. $\endgroup$ – David Richerby Nov 25 '14 at 23:23
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    $\begingroup$ $\{n=2k+1\,|\, k \in \mathbb{N}\}$ are the odd natural numbers (including $0$). So $2k+1$ is a sloppy version of indicating the property odd number. Used in $A_n$ vs $A_{2k+1}$. $\endgroup$ – mvw Nov 25 '14 at 23:48
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    $\begingroup$ But if you write $A_{2k+1}$, the reader is going to start asking questions. "Did I miss something?" "What's $k$?" "$k$'s a natural number? An integer?" "How is $A_3$ different from $A_5$ and $A_7$?" Sure, for any natural number $k$, $2k+1$ is odd but 38471 is odd. Why not just use $A_{38471}$? $\endgroup$ – David Richerby Nov 26 '14 at 0:26
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    $\begingroup$ Along that line of thought I would prefer $A_{13579}$ as symbolic substitution. :-) $\endgroup$ – mvw Nov 26 '14 at 9:29
  • $\begingroup$ @DavidRicherby A variation is $A_{2\mathbb{Z}-1}$ and $A_{2\mathbb{Z}}$. $\endgroup$ – Jeppe Stig Nielsen Nov 26 '14 at 12:11
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How about $A_\mathcal O$ and $A_\mathcal E$?

To produce these: A_\mathcal O and A_\mathcal E

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    $\begingroup$ @Jean-ClaudeArbaut It's not unheard of, though. Some excerpts from other papers: "Let ${\mathcal O}$ denote the family of all odd cycles ..."; "The Odd Distance Graph, denoted ${\mathcal O}$, ..."; "We let ${\mathcal O}$ denote partitions into odd parts ..."; etc. $\endgroup$ – Théophile Nov 25 '14 at 16:21
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    $\begingroup$ I wasn't aware of these, thank you, and +1 then $\endgroup$ – Jean-Claude Arbaut Nov 25 '14 at 16:36
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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.

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  • $\begingroup$ I like $A_0$ and $A_1$. Its nice. $\endgroup$ – goblin GONE Nov 25 '14 at 20:54
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    $\begingroup$ @goblin But $A_0$ could easily be construed as meaning "the $0$th element of $A$"—especially because $A$ is a sequence, not a set. $\endgroup$ – wchargin Nov 26 '14 at 3:45
  • $\begingroup$ @WChargin, true. So maybe $A_e$ and $A_o$ are better. $\endgroup$ – goblin GONE Nov 26 '14 at 4:19
  • $\begingroup$ @WChargin I agree that that's a potential problem. I think it would be OK if the sequence has been defined as $A = \langle a_1, \dots, a_k\rangle$ rather than $\langle A_1, \dots, A_k\rangle$ but if the sequence begins at $a_0$, the reader might wonder if $A_0$ and $A_1$ are typos for $a_0$ and $a_1$. (And I think I didn't notice the problem while I was writing the answer because the sequence in the question is defined as $\langle a,b,c,d,e,f\rangle$.) $\endgroup$ – David Richerby Nov 26 '14 at 10:20
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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.

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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.

Source

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