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June 2nd, 2015, 12:44 PM   #1
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Sequence with every integer in it

Hello,
I am not a maths expert, but rather a high school student who is really interested in this subject (and also not a native English speaker, thus sorry for mistakes). I was watching a video about sequence which has every integer in it, but it was built of two smaller sequences (one for 0 and positive integers, the other for the negative integers) with conditions when n is odd or even. But I was curious whether there exists a SINGLE sequence (without any additional conditions) which would have every integer in it, including zero. Using some trigonometric functions, I managed to build a single sequence which has members in it: 1, -1, 2, -2, 3, -3..., but I still lack 0 in it. My sequence is built so that nth member is defined by the index n.

I could not find any information on Google, thus I want to ask you: does there exist a single sequence with all integers in it, including zero?

EDIT: OK, I managed to get a sequence with all integers in it, including zero Anyway, if you know any sequence which would have all the the ingers in it, I would highly appreciate if you shared it. It would be curious to compare it to the one I created, because my expression is really "hairy" and so it is interesting whether it is possible to get a simplified expression.

Last edited by Yoshke; June 2nd, 2015 at 01:15 PM.
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June 2nd, 2015, 01:36 PM   #2
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You can put 0 wherever you like. For example:

0, 1, -1, 2, -2 ...

or

1, 2, 3, -1, -2, -3, 0, 4, -4, ...

I think you're confusing the existence of a sequence with the defintion of that sequence by a single formula. A sequence is a sequence however you define it.

You can define the first of these by:

(when n is odd)
(when n is even)

And that is perfectly well defined sequence of all the integers.
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June 2nd, 2015, 01:49 PM   #3
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Yes, I mean definition of a sequence by a formula. Your example is the one that I saw in the video, but as I said I am interested in a single formula defining the sequence without any additional conditions (odd, even index, for instance). I managed to get such formula for Nth term, including some simple trigonometric formulas, but I am curious whether there is a simpler way and I am surprised that I cannot find any information on the net.

By the way, my sequence defined by the formula starts 0, 1, - 1, 2, - 2, 3, - 3...
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June 2nd, 2015, 02:04 PM   #4
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Thanks from Yoshke
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June 2nd, 2015, 02:15 PM   #5
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Great. Thanks a lot.
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