The center of math retweeted the following problem: enter image description here

I surmised the answer is 22, using the following reasoning:

Odd entries increase by 2, whereas even entries increase by 1

$a_{1}=16, \, a_{2}=17, \, \dotso$

All seemed fine until I posed this question to a mathematician, who claimed this was not a mathematical sequence. Or at least it is not known to the mathematical community, i.e.

enter image description here

So, I am led to ask is this a sequence? Whether the answer turns out to be yes or no, I am more interested in the argument. It would be most helpful if there was some truth from the answer, which I could take away for future endeavors in the mathematical world.

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    $\begingroup$ Kind of strange to say that something isn't a sequence just because it doesn't match any sequence in the oeis. Especially since there are uncountably many integer sequence, so expecting oeis to have an entry for each of them seems like a lot to ask. $\endgroup$ – user2520938 Jan 15 '15 at 18:59
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    $\begingroup$ @user2520938: You are a master of meiosis. $\endgroup$ – Brian M. Scott Jan 15 '15 at 19:01
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    $\begingroup$ There is no inherently right answer to "what is the next number" questions. Whether it's in OEIS or not doesn't mean anything. $\endgroup$ – Nate Eldredge Jan 15 '15 at 19:04
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    $\begingroup$ @Hurkyl: Life is full of surprises. I guess that the title is just very objectionable. $\endgroup$ – Asaf Karagila Jan 15 '15 at 19:11
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    $\begingroup$ @Nate: I think they mean that the sequence is the interleaving of two sequences, one in which entries increase by 2 and one in which entries increase by 1. $\endgroup$ – user856 Jan 16 '15 at 4:13

A sequence is a mapping whose domain is a subset of the naturals.

So define $a: 0 \mapsto 16, 1 \mapsto 17, 2 \mapsto 18, 3 \mapsto 18, 4 \mapsto 20, 5 \mapsto 19$

There's your sequence. As to what the next number is, there's no way to know. I could say that the sequence continues as follows:

$a_n = 0$ for $n \ge 6$

There's no rule that a sequence has to follow any predictable pattern. And even if you figure out one pattern, the person who constructed these problems could have had another pattern in mind.

That's why these pattern-finding problems are at best not mathematics, and are at worse stupid.

Note that the sequence could be:

$a_n = \displaystyle -\frac {7}{90}n^5 + \frac {11}{9} n^4 - \frac {62}9 n^3 + \frac {148} 9 n^2 - \frac {401}{30} n + 16$

In which case the next term would be $- \dfrac {22}3$

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    $\begingroup$ I must disagree that these sort of problems are "stupid". They are ill-defined to some extent, perhaps. But there is often a solution that is simpler than others. If you think that the sequence 1,2,3,4,5,... should be followed by 78, -39, 23 then you are being obstinate. Occam's razor applies here; it is clear that the spirit of the question is to find a simple pattern. $\endgroup$ – Jair Taylor Jan 16 '15 at 2:57
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    $\begingroup$ @Jair: the stupidity is a function of how carefully the problems are formulated, right? people who ask "what's the next term" without explaining the silliness of the question, or asking exactly what they meant ("can you find a 'simple' pattern?"), are being dishonest. $\endgroup$ – symplectomorphic Jan 16 '15 at 3:08
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    $\begingroup$ @JairTaylor I was precise with my wording, I didn't say they're unequivocally stupid. Finding a pattern is inductive reasoning, which is an important skill. But it isn't math. $\endgroup$ – GFauxPas Jan 16 '15 at 3:45
  • $\begingroup$ @GFauxPas Well, I will agree with you that the question isn't mathematical in the strictest sense of that word. But I still think that answering "it could be anything!" without admitting that some answers are better than others is not helpful. $\endgroup$ – Jair Taylor Jan 16 '15 at 4:08
  • $\begingroup$ @symplectomorphic: The goal is pretty clear from the context here. I don't think it's really necessary to clarify this every time, although it might help. $\endgroup$ – Jair Taylor Jan 16 '15 at 4:12

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