Let's define $t_n$ by the recurrence $$t_0 = 1, \quad t_n = (-1)^n \, t_{\lfloor n/2\rfloor}.\tag1$$ It is easy to see that $|t_n|=1$, and the signs follow the same pattern as  the : $$1,\,-1,\,-1,\,1,\,-1,\,1,\,1,\,-1,\,-1,\,1,\,1,\,-1,\,1,\,-1,\,-1,\,1,\,...\tag2$$ (see  for an example of a non-recursive formula for $t_n$).  Now, define: $$\mathcal{S}(m, n)=\sum_{k=0}^{2^n-1}t_k\,k^m, \quad \color{gray}{m,\,n\in\mathbb{Z}^+.}\tag3$$ Computing some values of this sum immediately leads to the following conjecture:     $$\text{For all }\, n&gt;m,\,\,\mathcal{S}(m, n)=0.\tag{$\diamond$}$$   (e.g. for $m=2$ and $n=3$ we have $0-1-4+9-16+25+36-49=0$).  How can we prove it? Is it a known result?
There is a question I have based on the fact: If you take a quadratic polynomial with integer coefficients, take the set $\{1,2,3,4,5,6,7,8\}$, make a partition $A=\{1,4,6,7\}$, $B=\{2,3,5,8\}$, and then evaluate the polynomial with the elements of each set you got an equality.  If now with a cubic polynomial and the set $\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16\}$, how to pick a suitable partition that accomplish the same fact? I mean is there a general formula for bigger degree?
If I call a command using  in Ruby, how do I get its output?  system("ls") I have 2 graphics cards on my laptop. One is IGP and another discrete.   I've written a shell script to to turn off the discrete graphics card.  How can I convert it to systemd script to run it at start-up? After reading of:  I wonder if the  also applies to flagging of old questions.  I have the subjective feeling the longer the time between the question and the flagging is, the longer it takes to become reviewed.   Is flagging of old posts a waste of moderator time? Should I avoid that? Objective: To separate text column from figures column in every page I don't know if this has been asked before. But so far nothing has come up with my search.  In some books, the figures are placed at the outer side of the page like in the picture below.    Is this achievable in TeX/LaTeX? That is how does one set up the page so that all text will appear in a 'text column' and the figures appear in a 'figures column' in every page? Deleting polygon defined area from raster layer in ArcGIS Desktop? Which tool do I use in ArcGIS 10.1 for Desktop to exclude areas covered by vector polygon layers from a raster layer?   I want to keep portions of the raster layers that exclude the portions covered by the vector layer. Identifying a short story about time travel and paradoxes Most of something is plural or singular? I am writing a literature review. My question is which sentence is correct?   Most of the existing work focus on... Most of the existing work focuses on ... Most of the existing studies focus on ...   Thanks Extracting phone number from string Recent feature changes to Stack Exchange This is an unofficial list/changelog of new features and various changes to Stack Overflow and the Stack Exchange network.  It is maintained by the community, while a Stack Exchange employee changes the accepted answer to ensure that the latest changes remain on top (given default user settings).  To see the list in reverse chronological order (latest entry on top), .  For those adding new features here: please post only feature changes that affect the whole network and have some global impact. i.e. bug fixes don't belong here, nor do mobile app updates. Declare variable to hold job name and if job running kill it I’m looking for a single file that does these things in order.   Declare a variable to hold a job name  Run a query to check whether that job declared is currently running Set up an if loop If the job is running, execute the stored procedure that kills it.   My question is to write a query that declare a variable to hold a job name and gives us result the job is still running or not if the job is running execute the stored proc to kill it.  My manager was asking me to listing running jobs and suspended jobs and declare a variable to hold job name and setup an IF loop like if job is running run proc to kill it. I want to try to convert a string to a Guid, but I don't want to rely on catching exceptions (   for performance reasons - exceptions are expensive for usability reasons - the debugger pops up  for design reasons - the expected is not exceptional   In other words the code:  public static Boolean TryStrToGuid(String s, out Guid value) {     try     {         value = new Guid(s);         return true;     }     catch (FormatException)     {         value = Guid.Empty;         return false;     } }   is not suitable.  I would try using RegEx, but since the guid can be parenthesis wrapped, brace wrapped, none wrapped, makes it hard.   Additionally, I thought certain Guid values are invalid(?)    Update 1   had a good idea to catch only FormatException, rather than all. Changed the question's code sample to include suggestion.    Update 2  Why worry about thrown exceptions? Am I really expecting invalid GUIDs all that often?   The answer is yes. That is why I am using TryStrToGuid - I am expecting bad data.  Example 1 . I might be parsing folder names, checking to see if the text after the final . is a GUID.  c:\Program Files c:\Program Files.old c:\Users c:\Users.old c:\UserManager.{CE7F5AA5-6832-43FE-BAE1-80D14CD8F666} c:\Windows c:\Windows.old   Example 2 I might be running a heavily used web-server wants to check the validity of some posted back data. I don't want invalid data tying up resources 2-3 orders of magnitude higher than it needs to be.  Example 3 I might be parsing a search expression entered by a user.     If they enter GUID's I want to process them specially (such as specifically searching for that object, or highlight and format that specific search term in the response text.)    Update 3 - Performance benchmarks  Test converting 10,000 good Guids, and 10,000 bad Guids.  Catch FormatException:    10,000 good:     63,668 ticks    10,000 bad:   6,435,609 ticks  Regex Pre-Screen with try-catch:    10,000 good:    637,633 ticks    10,000 bad:     717,894 ticks  COM Interop CLSIDFromString    10,000 good:    126,120 ticks    10,000 bad:      23,134 ticks   p.s. I shouldn't have to justify a question. Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $ I just received a PhD invitation letter from a German University which mentions that my remuneration will be 2/3 of full TV-L 13 position. I do not know what a TV-L 13 position is (Google is of no help here), and didn't want to be rude, so I am asking here first. Is this a common position or specific to Germany? If anyone knows, currently how much remuneration corresponds to a TV-L position? Why was Arcanine with the Legendary Birds in Veridian City in Pokémon Origins? In Pokémon Origins, Red sees a image with the three legendary birds and Arcanine.     Why was Arcanine there? I mean, he's 'The Legendary Pokemon' but not a legendary Pokémon. In , there is a comment by  on the answer that Arcanine was originally a legendary Pokémon:     "...Nintendo first gave Zapdos, Articuno and Arcanine as the legendary trio. But they removed Arcanine as it would have been awkward. Two birds and a dog, huh!..."   This still does not add up because Pokémon Origins was created a while after they removed Arcanine. Any thoughts? Creating polygons as % of original area using ArcGIS Desktop? I have a polygon shapefile and need to create new polygons, within the originals, as % of original area.  I used buffer by field (Field= new area as %) but the buffers are created in circles around the polygons.  What am I doing wrong?  I am using ArcGIS Desktop 10. Flair: 500 Internal Server Error - http://stackexchange.com/error?aspxerrorpath=/users/flair/378431.png MicroSD card is set to Read-only state. How can I write data on it? I have a MicroSD card reader in my notebook. When I insert a MicroSD memory card into it, I can't write any data on it - it's read-only. The same thing happens when I try to do it under root. And I can't change the permission policy for it. This problem happens for all microSD cards (I've tested 2 cards). I've encountered it on Ubuntu 12.04 LTS, now I'm on Ubuntu 12.10, the problem is still on it's place. When I try to insert such a "read-only" card in my mobile phone, everything's OK and I can write data on it. So, can anybody tell me, how can I write a data onto the card?  UPD: here's fdisk -l output:  Disk /dev/sdc1: 2013 MB, 2013135360 bytes 62 heads, 62 sectors/track, 1022 cylinders, total 3931905 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disk identifier: 0x00000000   Device Boot      Start         End      Blocks   Id  System   The file system of the card is msdos. As to me it's weird.   UPD2: I've found a small "lock" switch on my card adapter, and it turns out, that when I insert it in my notebook, it switches on. Perhaps, I have to find some glue or something to get that switch stuck...  UPD3: There was no glue near me, so I've decided to buy another card-reader. How do you determine the winner of a hand? In Texas Hold'em poker, there are some cases where a particular card, often called kicker, acts as a tiebreaker between players to determine who wins the pot, or if the pot has to be shared.  For some combinations such as three or four of a kind, there is no doubt: the kicker is the 5th card of the chosen combination, and determines the winner.  Example     Alice has 7 and 10   Bob has 7 and K   On the board are 7, 7, 2, 3, 4   In that case, both have a three of a kind. But because Bob has a king and Alice only a 10, he wins.  Most of poker websites explaining the rules clearly mention the role of the kicker applying to three of a kind and double pairs. But much fewer say something about colors, straights and full houses, and I found many contradicting answers. I also asked the question to some used online poker players and they also gave me contradicting answers.  I have tried to search for official tournament rules, but most of them only explain what happens with bad behaviors, bad deals, incorrect or confusing betting, showing cards when you shouldn't, acting when it's not your turn, etc. without mentioning subtleties about combinations at all.  To simplify my question, I will take three examples; I think it's better to start with examples before getting to the general answer if one exists. So, what's happening in the following 3 examples? Could you give a more generalized answer?  Example 1 - Flushes     Alice has 2&clubs; and 3&clubs;   Bob has 4&clubs; and 5&clubs;   On the board are 6&clubs;, 8&clubs;, 10&clubs;, 2&hearts;, 5&diams;   Contradicting answer 1: it's a tie, because the highest card included in the flush is the 10, which everybody chooses to include in their 5 showdown cards. Contradicting answer 2: Bob wins, because he has the greatest private card that is part of the flush  Example 2 - Straight     Alice has 6 and K   Bob has 6 and J   On the board are 4, 5, 7, 8, 10   Contradicting answer 1: it's a tie, because the greatest card in the straight is the 8 for both Alice and Bob Contradicting answer 2: Alice wins, because she owns an extra king, compared to the jack of Bob  Example 3 - Full house     Alice has 3 and 7   Bob has 3 and 6   On the board are 3, 3, 2, 2, 5   Contradicting answer 1: it's a tie, because one is supposed to choose only five cards to make a combination, and a full house is already five cards. There couldn't be any kicker, and thus their showdowns are strictly equals. Contradicting answer 2: Alice wins because of her extra 7, compared to the 6 of Bob  Thank you very much for your answers.  Note: I'm unable to post next to you, so I edit my own post; strange not be able to answer to an answer.  Ok, So if I summarize what you are saying :   In the flush case, Bob wins because at some point, their hand differs. Technically, I can continue comparing the highest private card to decide who wins. I had it correct. In the straight case, if the highest card of the straight is public, then it's technically always a tie, no matter what the players had as second private card (asuming that only one of the two cards was part of the straight). In a full house if both players have the same triplet and the same pair, it's always a tie, no matter what the players had as second private card (assuming again that only one of the two was part of the full house).   Please confirm me that what I summarized is correct.  Thank you for your answer. Print LinkedList Recursively using C++ How to differ sessions in browser-tabs?