The problem of $k$ element subsets of an element set such that no two elements are consecutive can be solved using stars and bars approach. But, can it be also solved using recurrence relations and generating functions? Note that the problem of any element subsets can be solved using recurrences which gives a fibonacci number answer. But what about this problem? I think this is not duplicate problem. Any hints? Thanks beforehand.
I am trying to find the formula for the number $f(n,k)$ which is defined as the number of $k$-subsets of $[n]$ in which no two elements are consecutive numbers in $[n]$.  From the simple thinking I had made up following recursion:  $$f(n,k) = f(n-1,k) + f(n-2, k-1) $$  This is true since I can construct without last element of $[n]$ then make up $k$-subsets, then otherwise must contain the last element of $[n]$ then construct remaining $k-1$ elements among $[n-2]$ since with the last one, I can't use $n-1$.  However, I had never learned any techniques regarding changing those recursion into a well-summarized formula in terms of $n$ and $k$.  Any guidance would be appreciate.
Change date and time format from csv file without using date -d I have a .csv file contains   Data1|Data2|10/24/2017 8:10:00 AM   I want to change date and time format of column 3 as following:  From 10/24/2017 8:10:00 AM (12-Hour) to 20171024 08:10:00(24-hour).  Not using -d the_excerpt not displaying Forward/Inverse Search Does Not work on certain tex files What's the negation of "I used to be"? Surely not "I didn't used to be"? Wifi Automatically Turning ON after iOS 11 Update I'm facing this issue after updating iOS11 in my recently bought iPhone 7  I usually turn off wifi during the night and found it ON in the morning. It automatically turns ON resulting in unnecessary battery drains.  Would appreciate a fix !! I'm curious to know how important the bias node is for the effectiveness of modern neural networks. I can easily understand that it can be important in a shallow network with only a few input variables. However, modern neural nets such as in deep learning often have a large number of input variables to decide whether a certain neuron is triggered. Would simply removing them from, e.g., LeNet5 or ImageNet have any real impact at all? Verify the identity: $\tan^{-1} x +\tan^{-1} (1/x) = \pi /2$ Verify the identity: $\tan^{-1} x + \tan^{-1} (1/x) = \frac\pi 2, x &gt; 0$  $$\alpha= \tan^{-1} x$$  $$\beta = \tan^{-1} (1/x)$$  $$\tan \alpha = x$$  $$\tan \beta = 1/x$$  $$\tan^{-1}[\tan(\alpha + \beta)]$$  $$\tan^{-1}\left [{\tan\alpha + \tan\beta\over 1 - \tan\alpha \tan\beta} \right]$$  $$\tan^{-1}\left[ {x + 1/x\over 1- x/x }\right]$$  $$\tan^{-1}\left[{x + (1/x)\over 0} \right]$$  I can't find out what I'm doing wrong.. Can NDSolve handle discontinuous data? Is the electric force a vector or a vector field? How can I test for POSIX compliance of shell scripts? Considering that POSIX is the closest thing to a common standard among all unices, I'm interested in knowing if there's a shell that supports it exclusively. While most modern shells provide support for POSIX (and will run POSIX compliant scripts without any problem), they don't do a good job at pointing out non compliant features.  Is there any shell that implements POSIX and POSIX only, in such a way that it'd throw an error for any non compliant feature?  EDIT I want to clarify that I'm not asking for general tips for writing portable shell scripts. The related question mentionned in the comments already covered this. I thought of this question when I found out that bash has a --posix option but only to discover that it only affects some intialization behaviors which is not exactly what I'm looking for. Potato Power. Self sustaining medium using a living potato plant possible? When using sudo powers and calling gedit the top level menu with File Edit View Search Tools Documents Help is missing.  If I create user ID root and sign in once and call gedit will it create the necessary user configuration files that might fix this?  Will this also make the annoying error messages go away whenever I use sudo, gksu or pkexec as a regular user with elevated privileges togedit ?  Will there be other benefits with Nautilus and other Gnome derivative Ubuntu applications?  NOTE: by sudo powers you can google pkexec gedit 3.5k hits, gksu gedit 40k hits or sudo gedit 500k hits. I'm in the minority using the first method but I believe it will become the standard in Ubuntu 17.04.    June 12, 2017 Update Here are a list of errors you get when using pkexec gedit:  (gedit:13003): Gtk-WARNING **: Calling Inhibit failed: GDBus.Error:org.freedesktop.DBus.Error.ServiceUnknown: The name org.gnome.SessionManager was not provided by any .service files  ** (gedit:13003): WARNING **: Set document metadata failed: Setting attribute metadata::gedit-spell-enabled not supported  ** (gedit:13003): WARNING **: Set document metadata failed: Setting attribute metadata::gedit-encoding not supported  ** (gedit:13003): WARNING **: Set document metadata failed: Setting attribute metadata::gedit-position not supported   On this  there is a commitment to fix these distracting messages as of June 5, 2017. No indication is given for when the fix will be upstream nor if the functionality will be implemented or the error messages will simply disappear.  Note the bug report was filed over a year ago and over that time it only effects 18 people. How to best encapsulate monetary values in a database How to determine if a type implements a specific generic interface type Is the localization of a PID a PID? Let $A$ be a commutative ring and $S$ a multiplicative closed subset of $A$. If $A$ is a PID, show that $S^{-1}A$ is a PID.   I've taken an ideal $I$ of $S^{-1}A$ and I've tried to see that is generated by one element; the ideal $I$ has the form $S^{-1}J$ with $J$ an ideal of $A$. $J$ is generated by one element but I can't see why $I$ has to be generated by one element, maybe I'm wrong. The  is minimal and clean on a fresh installation of Windows. Install a bunch of applications and soon the context menu is loaded with all kinds of opening options from various applications.  How do I remove items from the right click (context) menu?  I find that there are different types of right click menu items:   Global items that appear in all context menus. Items that appear only on folders. Items that appear only on files. Items that appear only on special folders (Ex: Right clicking a folder of MP3s shows up a context menu with items like Play with Windows Media Player.) Items that appear only on certain file types (Ex: Right clicking a MP3 file shows up a context menu with items from Windows Media Player/Foobar2000/VLC/your-favorite-media-player begging to open this file.)   I want to be able to delete all these kinds items from the right click (context) menu. Error: The processing instruction target matching "[xX][mM][lL]" is not allowed Random test data: How to do it right? Since asking for tutorials, videos, resources, links to other content or downloads is considered off topic here, this is a specifically created Community Wiki which gathers resources for Blender and it has been approved by the .  Just write in the appropriate answer/section. If you have concerns, questions, post a  question, so we don't clutter the comments, but you can link your meta question from the comments.  Questions regarding such resources are not allowed anymore, except for very specific and on topic requests (ask on  if you're unsure about your question). Follow the instructions made in the question about how to post, what can be posted, etc.  Learning             Assets    (also see: )   Reference Images/Blueprints (also see: )    Tools    (also see: )      Other   Special Interest ()     Also see the . How to get a list of applications installed in Ubuntu dash? In Gnome, we have applications grouped up based on their type (like - games, accessories, et al). But in Unity, we need to manually type in the Unity to get the App.  Is there any keyword/utility to display the applications in GUI?  Ubuntu Version : 14.04 Hardware : Dell Inspiron 3437