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If $F_n$ is the $n$-th fibonacci number, then prove that, $$F_n={n-1 \choose 0 }+{n-2 \choose 1 }+{n-3 \choose 2 }+\ldots$$  I tried the idea of using Pascal's triangle, but it seems to need some adjustment, so I am stuck, please help.    Any combinatorial argument would be of greater help than rigorous proof.  Thank you. Why is the $S_{z} =0$ state forbidden for photons? I am looking for an animated series, shown in the 90s. I remember an episode where the boy went into VR and fought a medusa-like monster.  It may be that this was the premise for the whole series: That the children heroes go into VR every time to fight the evil guys.   I am not sure if the whole series were computer-animated but surely there were such elements together with classic animation.  The style of the series was quite dark and cool for teenage boys. On what site can I ask questions related to copyright? I want to make a summary of a book and use it to earn money. However, I don't know whether it's legal or not.  Which Stack Exchange site can I ask this question on? What is the best way to restrict access to adult content? I bought my kids a PC and installed 12.04 (Unity) on it.  The bottom line is, I want my children to use the computer unsupervised while I have confidence they cannot access anything inappropriate.  What I have looked at:  I was looking at Scrubit a tool which allows me configure my wifi router to block content and this solution would also protect my other PC and mobile devices.  This may be overkill as I just want the solution to work on one PC.    I also did some Google searches and came across the application called .  My experience of OSS is that the best solutions frequently never appear first on a Google search list and in this case I need to trust the methods therefore my question is very specific.   I want to leverage your knowledge and experience to understand What is the best way to restrict adult content on 12.04 LTS as this is important to me.  It maybe a combination of things so please don't answer this question "try this or that", then give me some PPA unless you can share your experience of how good it is and of course if there are any constraints. Dealing with bundle/omnibus/list-of-questions question that consists of rather distinct questions I have looked at the various help pages but I couldn't find a clear guideline for the following issue: often enough someone will ask in the same post (i.e. in the same SE-question) a whole bunch of questions (in the plain English sense of the word), which while may be about the same topic are nevertheless sufficiently distinct that each may require several paragraphs of text to answer adequately.   I can see an upside for leaving related questions grouped together from the asker's perspective: "there's a thread (on SE) answering all my questions about topic T!" On the other hand, from the perspective of organizing questions by having a clear heading/title for each (actual) question and so forth, i.e. from the perspective of SE being a fairly structured database of Q&amp;As instead of being a more like a collection of forum threads "I'm confused about topic T", the omnibus question-list approach seems clearly sub-optimal.  The omnibus approach also makes the issue of accepting answers (and perhaps reviewing such actions) difficult: if two different users give replies to different parts/sub-sets of the question-list, which one is the most adequate one? Even upvoting all partial answers seems iffy because it actually penalizes the user(s) who provides(s) the most complete answer to the question-list: since a user can only give at most one upvote to another user's answer, if user X provides answers to A &amp; B in the same SE-answer, he can only get at most one upvote from user Y, whereas if these were two different SE-answers, he could get two upvotes from user Y.  So, given the above problems, which I think are real enough, is there anything in the SE system addressing list-of-questions question splitting? How do they decide Blu-ray/DVD release date? I came across a list of functions in my calculus textbook whose indefinite integral cannot be found. It was written that the integral $$\int \frac{\sin(x)}{x} dx$$ cannot be evaluated without any explanation as to why.  I did some research over the internet and found out that the definite integral $$\int_{0}^{\infty} \frac{\sin(x)}{x} dx$$ can be evaluated using Laplace transformation and is equal to $\pi /2$. But I still couldn't find answer to my original question. I read somewhere that the integral cannot be expressed using 'elementary functions'. A little help is appreciated, I'm in Calc 1 going advanced than my course but I am sorry if my post shows lack of research. Thank you! Problem on Extremal Graphs: three independent paths The problem states that:     A simple (or strict) graph with $n$ vertices and $m&gt;\dfrac{3(n-1)}{2}$ edges, has two vertex   joined by three independent paths.   Any hints, ideas or useful results, to attack this problem? Keyword for the outer class from an anonymous inner class In the following snippet:  public class a {     public void otherMethod(){}     public void doStuff(String str, InnerClass b){}     public void method(a){         doStuff("asd",             new InnerClass(){                 public void innerMethod(){                     otherMethod();                 }             }         );     } }   Is there a keyword to refer to the outer class from the inner class?  Basically what I want to do is outer.otherMethod(), or something of the like, but can't seem to find anything. I flagged a post today, typed out a fairly thorough explanation, clicked the button... and got told that I'd already flagged this post (for something else, which I'd forgotten about) so this flag wasn't allowed.  (This happens if the first flag is still active.)  The dialogue with my explanation was gone at this point, so if I wanted to communicate with the moderators about that, I was going to have to start over.  (In case you're wondering how this could happen, one flag was related to the post itself and one was related to the user -- there being no way to flag a user directly, you have to hang it off of a post.  But I've also seen this with cases where the first flag hangs around unhandled for a week or two and I've forgotten about it.)  Since the system is going to detect that I've already flagged a post and reject a second flag, can it do that after I click the "flag" link instead of waiting until the end of the process?  (Or gray out the link so I can't click it in the first place?)  2015-12-18: someone suggested to me in chat today that this had been implemented, so I tested again.  The behavior has not changed: I flagged a post using "other", then flagged it again (also "other") before the first flag was handled, and the flag was rejected only after I'd typed the message.  (I did this on a site where I'm a moderator so I could then clean up after myself.) Can't pair apple wireless keyboard Keyboard: Apple Wireless Keyboard (model number MC184LL/B)  System: Macbook Pro Early 2011 running OS X 10.8  The keyboard gets detected and when I try to pair it, the system asks to enter some numbers through the keyboard.   However when I press the key (on the wireless keyboard) nothing happens, the only key that seems to work (i.e. to move to the next key in the code) is 0. I can also press enter to confirm but it always fails (since the only code I can enter is 0000...0).   I've also tried pairing from the bluetooth preference and changing the passcode option. However this won't work either.   Any ideas? The keyboard was paired with another MacBook Pro (I don't have access to it anymore). Is there any method to 'reset' the keyboard to factory settings? Closed form for $\int_0^\infty\frac{1}{(1+x^2)^s}\,dx$ when $s\in (0.5,\infty)\setminus\mathbb{N}$ I know that the improper integral  $$ \int_0^\infty\frac{1}{(1+x^2)^s}\,dx $$ is convergent for $s&gt;0.5$ and divergent otherwise. Furthermore, it has a closed form for $s \in \mathbb{N}$ (this can be obtained using the Residue Theorem). My question is to know whether it has a closed form for any non-integer $s &gt;0.5$. Strip whitespace from jsp output Unable to install .NET 3.5 on Windows 8.1 I am trying to install .NET 3.5 to work on few legacy applications and this has been burning my head for past few weeks.   During installation a pop up window requests me to install .NET 3.5 and when i hit download it fails stating issues with internet connection. But internet works just fine Tried enabling .NET 3.5 from Control panel it redirects to " Download from Windows update " and the download goes on forever ( best try 6 hours on 18mbps wifi keeps loading..) Using Dism with and without /LimitAccess the process gets stuck at 65.8% and then throws a  Downloaded offline package 253 MB from microsoft when i tried to install it again asks me download 3.5 to launch dotfx35.exe which is the setup file :( Tried everything from  no luck   Please help.. :( Python: access class property from string Didn't Harry have any other family he could have searched for? How to prevent system applications (like the Software Center) from asking for password? Redirect all requests to subdirectory except a few ip addresses NPN circuit to switch 12v 1A DC load by 3.3v 1A DC using PIR