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 June 2nd, 2015, 12:44 PM #1 Newbie   Joined: Jun 2015 From: Lithuania Posts: 3 Thanks: 0 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.
 June 2nd, 2015, 01:36 PM #2 Senior Member   Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 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: $s_n= \frac{n -1}{2}$ (when n is odd) $s_n= -\frac{n}{2}$ (when n is even) And that is perfectly well defined sequence of all the integers.
 June 2nd, 2015, 01:49 PM #3 Newbie   Joined: Jun 2015 From: Lithuania Posts: 3 Thanks: 0 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...
 June 2nd, 2015, 02:04 PM #4 Senior Member   Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 $s_n= (-1)^{n+1}(\frac12)[n - \frac12 (1 + (-1)^{n+1})]$ Thanks from Yoshke
 June 2nd, 2015, 02:15 PM #5 Newbie   Joined: Jun 2015 From: Lithuania Posts: 3 Thanks: 0 Great. Thanks a lot.

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