If I let $n \geq 1$ be an integer and use a $2 \times n$ board $D_n$ containing  $2n$ cells, each side has a length of 1. T  The brick can be vetical or horizontal containg $2$  cells(explained in the bottom part of the above picture).  The tiling of board ($D_n$) is placed with bricks so that :  1.item the bricks exactly cover $D_n$ and  2.item no two bricks overlap.  So for $n \geq 1$, we let $x_n$ become number of different tilings of the  board $D_n$. Now I have to determine the value of $a_n$  I know there are only two ways to cover the left-most two squares: either a single vertical brick, or two horizontal bricks and that if you cover those two left ones with a single vertical brick, after that you have to choose any covering of the other $n-1$ columns and the only way to do this is precisely $x_{n-1}$. So, there are $x_{n-1}$ arrangements in which you cover the left two by a vertical brick and if you need two horizontal bricks, Then you've knocked out two columns, and need to pick a covering of the remaining $n-2$ columns. So now what would $x_n$ equal I think $a_n = a_(n-1) + a_(n-2)$ I am I right?
I'm looking to find out if there's any easy way to calculate the number of ways to tile a $3 \times 2n$ rectangle with dominoes. I was able to do it with the two codependent recurrences  f(0) = g(0) = 1 f(n) = f(n-1) + 2g(n-1) g(n) = f(n) + g(n-1)   where $f(n)$ is the actual answer and $g(n)$ is a helper function that represents the number of ways to tile a $3 \times 2n$ rectangle with two extra squares on the end (the same as a $3 \times 2n+1$ rectangle missing one square).  By combining these and doing some algebra, I was able to reduce this to  f(n) = 4f(n-1) - f(n-2)   which shows up as sequence , confirming that this is the correct recurrence.  The number of ways to tile a $2 \times n$ rectangle is the Fibonacci numbers because every rectangle ends with either a verticle domino or two horizontal ones, which gives the exact recurrence that Fibonacci numbers do. My question is, is there a similar simple explanation for this recurrence for tiling a $3 \times 2n$ rectangle?
There are now errors when updating and I cannot install most software due to a corrupted  /etc/apt/sources.list   file. Is there a copy I can download to replace it?  The file would be for Ubuntu 12.04 (Final Beta) in the United States. Getting the default text editor used in system I have used select-editor and I note that I am using /usr/bin/nano. Is this the default text editor I am using?  When I open text files, It opens with gedit  What command in terminal must be specified to get the default text editor that I am using, and have it returned to e in terminal? Where in the file system are applications installed? I have installed a few Android applications both with the adb install MyApplication.apk command and via a webserver (like an alternative to Android Market).  But where in the Android file system on my phone is the MyApplication.apk file placed? Are applications installed with adb install not placed in the same directory as applications downloaded from the Market?  I found some applications on /system/app but I can't recognize them as some downloaded from Android Market or the one I have installed with adb install.  Where on the system is applications that I have installed located?  My phone is not rooted, and I am running Nexus One with Android 2.2.1. What are the differences between "lay" and "lie"? Citing two authors gets displayed incorrectly Symbolizing using RGB values from Excel in ArcGIS Desktop? I have an excel file with 2 attribute columns, 1-feature info 2-Number of color (RGB) that the current feature should be symbolize. This file file can be joined with the layer using the "feature info" field which is unique (join by attribute table). So i have as attribute the color number attribute for every polygon feature.  How can  i use this this information to symbolize every feature with their attributed color already imported from the excel sheet? How to do this the fastest way? Is it possible to serialize and deserialize a class in C++? Should Suggested Edits change the technical content of a post? I have noticed a significant number of suggested edits that attempt to fix code, or change the original meaning of someone else's post.    Is this really a good idea?    I've been on StackOverflow awhile, and among the editors who have sufficient reputation to edit without approval, I've never seen anyone modify the technical content or meaning of someone else's post.   I would consider obvious syntax and spelling errors OK to change, but I have seen many suggested edits that change the meaning of the code that was posted, or add things to the code that have nothing to do with the original question. A good example of this would be adding parameters to a SQL command, when the question was not asking about SQL injection. How to install Flash player on Ubuntu? Fancy vertical timeline I'd like to have a timeline as similar as possible to this with the addition of being able to add a logo at the left of each textbullet, but I've got no clue at all:    Thanks very much in advance for any help :)  p.s. I've stumbled upon this answer () but I don't how to modify it to get what I want! Is there anything beyond subtle hints that Jack becomes the Face of Boe? About $\lim \left(1+\frac {x}{n}\right)^n$ I was wondering if it is possible to get a link to a rigorous proof that  $$\displaystyle \lim_{n\to\infty} \left(1+\frac {x}{n}\right)^n=\exp x$$ Why is $a^{-1}$ mod $p$ equal to $a^{p-2}$ mod $p$ in modular arithmetic? Seeking English term for relationship between two members of extended family? Windows page file on multiple hard drives I cant find any documentation or articles online online for this...  how does Windows handle page files on multiple hard drives?  Does windows treat them as a concatenation, writing to one till its full then moving to the next?  Or does it treat them like a stripe, writing to each one incrementally?  Does it write to the first one free?  Or perhaps some other method? Denote by $M_{n \times n}(k)$ the ring of $n$ by $n$ matrices with coefficients in the field $k$.  Then why does this ring not contain any two-sided ideal?    Thanks for any clarification, and this is an exercise from the notes of Commutative Algebra by Pete L Clark, of which I thought as simple but I cannot figure it out now. How to create a link to a URL provided in a bib file with custom text Using biblatex, how can I get a URL from the bib file so that I can use it in the \href command?  I am guessing that the \usefield command is what I am looking for and that in order to use that I would need the \entrydata command. But both are undefined.  \documentclass{article} \usepackage[backend=biber]{biblatex} \usepackage{hyperref}  \usepackage{filecontents} \begin{filecontents*}{sources.bib} @online{biblatex,     url = {http://mirrors.ctan.org/macros/latex/contrib/biblatex/doc/biblatex.pdf},     urldate = {2018-02-02}, } \end{filecontents*}  \addbibresource{sources.bib}  \begin{document}      % This would work, but I don't want to duplicate the URL from the bib file:     \href{http://mirrors.ctan.org/macros/latex/contrib/biblatex/doc/biblatex.pdf}{biblatex documentation}      % Does not work because \citeurl{biblatex} is already a link, not the URL:     %The \href{\citeurl{biblatex}}{biblatex documentation}      % Does not work because \usefield and \entrydata are undefined:     %The \entrydata{biblatex}{\usefield\href{url}{biblatex documentation}}      \printbibliography \end{document} Why was Sirius ever convicted? Sirius himself says at some point in Goblet of Fire that Crouch pushed through legislation to have him convicted and thrown in Azkaban without a trial.  There seems to be a general consensus, specifically voiced by Fudge but also implicit in Crouch pushing through this legislation, that Sirius was considered one of Voldemort’s innermost, most dangerous, and most trusted right hands. Easily as inner-circle as the Lestranges, for example.  It seems quite unlikely to me that someone so close to Voldemort should not be a Death Eater—apart from Fenrir Greyback (who I would think is much less inner-circle), I do not recall hearing of any of Voldemort’s important servants not being Death Eaters.  As far as canon describes, all Death Eaters have the Dark Mark magic-tattooed on their arm, to be used as a way for Voldemort to summon them to him. Considering that Karkaroff, who had definitely burnt all his Death Eater bridges, still had the Dark Mark, I am assuming that getting rid of the Mark is not possible: once it’s on, it stays on.  Yet, as Hermione points out in Order of the Phoenix (ch. 16, “In the Hog’s Head”), Sirius does not have the Dark Mark.  So why was he so swiftly convicted and locked up? I mean, I know they were eager to be seen to be doing something (as Fudge keeps pointing out is his excuse for most of the miscarriages of justice he commits)… but seriously, did no one think to check if Sirius actually had a Dark Mark on his arm or not? Did the Ministry of Magic not even know at the time that all Death Eaters had the Mark? Or were they just happy to accept—without the scrutiny of a full trial—that a non-Death Eater would commit mass murder for a Voldemort who was presumed defeated? Find $n$ such that $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$. I have to find the form of n i.e. whether n is even or odd and whether it is multiple of 2 or 3 such that:  $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$  What I tried:  $$x^2 + x + 1 = (x + 1)^2 - x $$  also,  $$(x+1)^n-x^n-1= \binom{n}{o}x^n + \binom{n}{1}x^{n-1} + ... + \binom{n}{n-1}x + \binom{n}{n}.1 - x^n - 1$$  $$=\binom{n}{1}x^{n-1} + ... + \binom{n}{n-1}x$$  after getting this far I tried with taking x and n common but still couldn't able to find the right direction.  I need extra hints to solve this.