Since childhood we are all familiar with the task of predicting the next number in a sequence. From something simple like, $2,4,6,...$ and $4,9,25,...$ to something more complex like, $0,1,2^2,4^2+1,(4^2+2)^2,(4^2+2)^2+1,((4^2+2)^2+2)^2$ which would look like this if it was asked as a question $0,1,4,5,36,37,1444$.  There (probably) is an infinite number of different possible patterns.  My question is, does an algorithm exist that can predict any type of pattern? And considering that I think that the answer is no, my other question is, are there any algorithms that can predict any pattern to a certain level of complexity?  EDIT  @Thomas Andrews Thank you for pointing out the flaw with my question. In light of that, I would like to change my question to this:  Is there an algorithm to determine a finite (or infinite) list of possible functions that match the given sequence?
I'm a trainee teacher and I need to pass a skills test in maths, but I'm having trouble in this particular area of algebra in finding the nth term of any sequence. I've been given a series of questions to answer one of them is:  Find the nth term of the sequence 3, 8, 15, 24.  I not only need to know the answer but more importantly the technique. This is GCSE level.
I would like to create a minecraft server for me and and my boyfriend to play on.  How do I set it up? Do I need any additional software? Do you have some links to walkthroughs or tutorials?  I have tried to set this up a few times but have failed every time. Find the derived subgroup of $A_4$ Find the derived subgroup of $A_4$.   Since it is $A_4$, for a permutation $\sigma$ to be in $A_4$, $\sigma$ must have a cycle structure of $2$ cycles. Therefore, $\sigma=(ab)(cd)$.  The commutator of such elements, would be obviously another permutation of $2$-cycle (of length $2$).  Does it mean that $A'_4=A_4$? "Its" as a Possessive Pronoun Best way to center a &lt;div&gt; element on a page both vertically and horizontally?  I know that margin-left: auto; margin-right: auto; will center on the horizontal, but what is the best way to do it vertically, too? How do I use one shape to delete a portion of another shape? I have created an arrow shape (rightmost in image), from which I wish to cut out the majority so that only two thin sides remain. The most accurate way to do this would be to duplicate the shape and to somehow use the duplicate (leftmost in image) to remove the chunk that I no longer desire from the original. I am not sure what to do after the duplication, however, or if this method is even possible? Is there a better one? When do you use POST and when do you use GET? Text processing - join every two lines with commas I have more than 1000 lines in a file. The file starts as follows (line numbers added):  Station Name Station Code A N DEV NAGAR ACND ABHAIPUR AHA ABOHAR ABS ABU ROAD ABR   I need to convert this to a file, with comma separated entries by joining every two lines. The final data should look like  Station Name,Station Code A N DEV NAGAR,ACND ABHAIPUR,AHA ABOHAR,ABS ABU ROAD,ABR ...   What I was trying was - trying to write a shell script and then echo them with comma in between. But I guess a simpler effective one-liner would do the job here may be in sed/awk.  Any ideas? $M$ connected $\iff$ $M$ and $\emptyset$ are the only subsets of $M$ open and closed at the same time I'm trying to understand this proof that:  $M$ connected $\iff$ $M$ and $\emptyset$ are the only subsets of $M$ open and closed at the same time  Which is:  If $M=A\cup B$ is a separation, then $A$ and $B$ are open and closed. Recriprocally, if $A\subset M$ is open and closed, then $M = A\cup(M-A)$. What? I know that if $M=A\cup B$ is a separation, $A$ and $B$ are both open. But why closed? Also, the 'recriprocally' part is totally nonsense to me. Anybody could help?  Also, there's another proof, which states: $M$ and $\emptyset$ are the only subsets of $M$ at the same time closed and open $\iff$ if $X\subset M$ has empty boundary, theb $X=M$ or $X=\emptyset$  which is proved as the following:  given $X\subset M$, we know the condition $X\cap \partial X = \emptyset$ implies $X$ is open, while the condition $\partial X \subset X$ implies $X$ is closed. Then, $X$ is open and closed $\iff$ $\partial X = X\cap \partial X = \emptyset$, this show $\iff$ for the theorem above.  First of all, I think that the condition $X$ has boundary empty implies that $X\cap \partial X$ is empty, but who said anything about $\partial X\subset X$? Also, where's the $\rightarrow$ of this proof? I can only see $\leftarrow$ Hot questions no longer work on iPad (ok on iPhone) I like to edit my answers. To clarify, illuminate, add examples, add links, point to new information as it crops up, etc.  It's highly irritating that my effort to improve my answer turns it into a Community Wiki.  Could anything be done to keep me from turning into a collective? What is the association bonus, and how does it work? Search for emails from contacts with multiple email addresses Is it possible to paginate posts correctly that are random ordered? I found this problem on Wordpress Support and the topic unfortunately is now closed.  I have this same issue... (read below)    We have created a site where members can recommend things like favourite books, movie, songs, etc. For this problem I will use the Movies page as an example.  The "Movies" page is ultimately a custom page template that asks wordpress to display a random list of ALL posts that have been given the category "movies" (category 31). It displays the title of these movies in random order using the code below.  &lt;?php  $rand = new WP_Query("cat=31&amp;showposts=-1&amp;orderby=rand");  while($rand-&gt;have_posts()) : $rand-&gt;the_post(); ?&gt;" rel="bookmark" title="Permanent Link to &lt;?php the_title_attribute(); ?&gt;"&gt;&lt;?php the_title(); ?&gt; &lt;?php endwhile; ?&gt;   Problem is the list is getting rather long and I would like to split it up into two or more pages of about 10 movies each. To achieve this, I have used the code below.  &lt;?php  $page = (get_query_var('paged')) ? get_query_var('paged') : 1;  query_posts("cat=31&amp;orderby=rand&amp;showposts=10&amp;paged=$page");  while ( have_posts() ) : the_post()  ?&gt;" rel="bookmark" title="Permanent Link to &lt;?php the_title_attribute(); ?&gt;"&gt;&lt;?php the_title(); ?&gt; &lt;?php endwhile; ?&gt;   But there is a problem because although it splits up the data into pages of 10 posts each (paginates), it fails to include a new set of 10 posts on page 2 and so forth. In other words, because it is listing things in a random order, it just goes out and gets another 10 random posts (or in this case movie titles). As a result, we have some repeated movie title posts instead of a new set of 10 random movie titles on page 2, etc.  My question is - what can I add to this code to get wordpress to "remember" which random 10 posts it included on page 1, and then have it get a new set of 10 posts to put on pages 2, 3, etc. until all posts are displayed. I would like there to be only one occurrence of a post per page when sorted randomly in sets of 10. Differences between strong and weak in Objective-C Accessing derived class property members from base class object in CSharp Convergence of $\sum_{n=1}^{\infty}{\sqrt[n]{n}-1 \over n}$ I am trying to conclude about the convergence of $$\sum_{n=1}^{\infty}{\sqrt[n]{n}-1 \over n}$$ If I take ${\sqrt[n]{{\sqrt[n]{n}-1 \over n}}}= {\sqrt[n]{\sqrt[n]{n}-1} \over \sqrt[n]{n}}$ this looks like it converges to $0$ but I am suspicious about the numerator. At the same time, other techniques I know like the ratio test, $n(\sqrt[n]{n}-1)$ and comparisons don't yield meaningful results. Any hints? There is an experimental timeline view for questions now available on meta ONLY! (Now working on SO, SF, and SU as well)  This is partially motivated by comments about how difficult it is to follow very active questions which have a lot of activity, and also the feedback on the API which was heavily dominated by "we need to know what happened in this time range" type responses.  It can only be accessed through URL hackery at the moment. To get to it, enter question URLs like so:  https://meta.stackexchange.com/posts/{id}/timeline  This is a chronological timeline of everything that happened on a question, including   initial question ask answers  revisions comments closure, deletion, reopen, undelete, etc votes   all based on when they happened in chronological order.  We are thinking about making this a fourth tab on the question, like so:  Timeline | Oldest | Newest | Votes  Please try it out and provide feedback. It is still a little rough, so we are not looking for highly detailed bug reports, just more general feedback.  Timeline view for this question: Primitive polynomials I am revising for a discrete mathematics exam and as quite stuck on this question.  Show that the polynomial $f = x^2 + 2 x + 3 \in \mathbb{Z}_5[x]$ is primitive. How many monic primitive quadratic polynomials are there in $\mathbb{Z}_5[x]$?  Any help would be greatly appreciated. This is the announcement post of the Stack Exchange Data Explorer. Its answers were meant to iron out the initial glitches 6 to 8 years ago.  If you have found a bug or need support  today with any of SEDEs features, please ask a new question and tag it with   What does it do? Allows you to run arbitrary SQL queries on the data dumps, and share those queries with your friends. Technical details It is hosted on the Stack Exchange infrastructure , the database is hosted on the same SQL Server version all other sites are hosted on. You can use most of the TSQL commands, even create temporary objects (but not User Defined Functions). The controls used are jQuery based, the grid is  . All query results are cached, so sharing a link to a query will not result in the server going in to a tailspin. The query edit control is a slightly modified  control, I also am considering bespin cause the inter iframe communication is nasty complicated. It gives up nice syntax highlighting for queries. Parameters There is support for parameterized queries, to get it to work name your params like ##this##. For example have a . Query name and description A query can have a name and description, to name it lead with a comment, to describe it continue commenting, for example: -- This is my query name  -- This is my description -- I can span multiple lines  SELECT 1  Anonymous users can initially name a query, but can not change the names of any existing queries. Tutorial There's a  which helps you getting started in writing your own SQL queries. Logging in You can use your openid provider to log in, at the moment once logged in we will track the list of queries you execute (giving you basic history support) Also if there are any params named UserId they will be populated automatically if your EmailHash matches the one on the site being queried. Magic columns If you alias an id column with as [Post Link] it will automatically create a link in the result set to the parent site. Similarly if you alias an id column with as [User Link] it will display a link to the user page. select top 10 * from Posts will show you how tags are done. (Magic columns for images is planned) example (most controversial posts:  ) So what are you waiting for, check it out at  Any feedback would be greatly appreciated. I hope to have the login section working fairly soon and allow voting on the best queries and commenting. Featured Queries At the moment, admins have permission to feature interesting queries on this page , I am looking for better ways to manage the huge query list that is building up. How can I keep low resolution textures pixelated?