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Significance of starting the Fibonacci sequence with 0, 1.... DISCLAIMER: I do not deal with in-depth mathematics on a daily basis as some of you may, so please pardon my ignorance or lack of coherence on this topic.  QUESTION: What is the significance of starting the Fibonacci sequence with $0,1$ ?  For instance, if I picked any two random integers, say 2 and 7, to start a sequence would I actually be creating some multiple or derivation of the Fibonacci sequence?  Is there a general mathematical explanation for the relationship between any sequence represented by $a[0] = x, a[1] = y, a[n] = a[n-1] + a[n-2]$ and the Fibonacci sequence?  Or, back to my example sequence, is there a general mathematical relationship between:  $2,7,9,16,25,41,66,107,173,280...$  and  $0,1,1,2,3,5,8,13,21,34...$  Perhaps the Golden Ratio explains it somehow?  Any help would be appreciated. Counting invertible $0$-$1$ matrices that remain invertible under negating the diagonal There is an invertible $n×n$ matrix with each entry $0$ or $1$. Now we negate ($x\to 1-x$) the diagonal of the matrix, and it is still invertible. How many such matrices exist for a given $n$?  Can we derive a general formula? If $n=2$ there are $2$ such matrices, if $n=3$ there are $48$. But I have no idea whether a formula exists for higher $n$. Best practices for storing UI settings? Remove Background in Inkscape How can light be called electromagnetic if it doesn't appear to be electric nor magnetic?  If I go out to the sunlight, magnets aren't affected (or don't seem to be).  And there is no transfer of electric charge/electrons (as there is in AC/DC current in space).   In particular, the  (which light is supposed to be composed of) have no electric charge (nor do they have ).   I'm looking for an explanation that can be appreciated by the average non-physicist Joe. Reference for the subgroup structure of $\rm{PSL}_2(q)$ Let $E_n$ be $$E_n:=\{f\in C[0,1]\mid \text{exist } x_f\in[0,1] \text{ such that } |f(x)-f(x_f)|\leq n|x-x_f|,\, \forall x\in[0,1]\}.$$  How show that $E_n$ is nowhere dense, that is, $\mathrm{int}\  \overline{E_n} = \emptyset$?  Can someone help me? "Which" vs. "what" — what's the difference and when should you use one or the other? Most of the time one or the other feels better, but every so often, "which" vs. "what" trips me up.   So, what's the exact difference and when should you use one or the other? Equivalence of conditions for Convergence of Non-negative Random Series Question  Let $X_n\geq 0$ be independent for $n\geq 1$. The following are equivalent.  $\sum_{n=1}^\infty X_n&lt;\infty$ a.s. $\sum_{n=1}^\infty[P(X_n&gt;1)+E(X_nI(X_n\leq 1)]&lt;\infty$ $\sum_{n=1}^\infty E(X_n/(1+X_n))&lt;\infty$   My attempt I was able to prove $(1)$ implies $(2)$ and $(2)$ implies $(3)$ but not $(3)$ implies $(1)$. For $(1)$ implies $(2)$, observe that by Borel-Cantelli $\sum_{n=1}^\infty P(X_n&gt;1)&lt;\infty$, else $X_n&gt;1$ infinitely often w.p.1 so that $(1)$ is violated. To see that $\sum_{n=1}^\infty E(X_nI(X_n\leq 1)]&lt;\infty$, apply Kolmogorov three-series theorem (I think). For $(2)$ implies $(3)$ we can write that $$ \begin{align} \frac{X_n}{1+X_n}&amp; =\frac{X_n}{1+X_n}I(X_n\leq 1)+\frac{X_n}{1+X_n}I(X_n&gt;1)\\ &amp;\leq X_nI(X_n\leq 1)+ I(X_n&gt;1). \end{align} $$ Now take the expectation of both sides and sum on $n$ using $(2)$ to conclude $(3)$. For the last implication $(3)$ implies $(1)$, I tried to use Kolmogorov three series theorem. To this end put $Y_n=X_nI(X_n\leq 1)$. First note that $\sum_{n=1}^\infty P(X_n&gt;1)&lt;\infty$ since $$ P(X_n&gt;1)\leq P\left(\frac{X_n}{1+X_n}&gt;\frac{1}{2}\right)\leq 2E\frac{X_n}{1+X_n} $$ by Markov's inequality. But I am unable to show that $\sum EY_n&lt;\infty$ and $\sum\text{Var}(Y_n)&lt;\infty$. Any help is appreciated. Here's what is happening; everytime I go to Stack Overflow (stackoverflow.com, NOT meta) with Firefox, it gives me the "Welcome back" tab that pops up from the top. Whenever I click in the "click here to refresh the page" link, all it does is refresh the page without logging me back in.  Even if I try to log in using the "log in" button at the top, all it does is pop up the tab that says "Welcome name here, you're being redirected." The problem is I'm not being logged in, not to mention that it keeps refreshing the page over and over again without giving me a choice to log in with Facebook.  For some reason, I was capable of logging in to Meta Stack Overflow using Firefox. I am still unable to log in to Stack Overflow. Any ideas? Automating "Stay Awake" I use my personal Android smartphone for development. Every morning, I go to Settings \ Developer Options \ Stay Awake and check the box.  At the end of the day, I uncheck the box — since otherwise any notification (including the "fully charged" one) will trigger the screen on, and it will stay on from that point on.  How can these two operations be automated? Short circuit evaluation and side effects Custom question lists: finding questions you can answer I'm a US citizen and I've already gotten my Tier 4 visa to study in the UK in my passport, however, it looks like the start date on the visa and the letter to present to the Immigration Officer has allowed me entry an extra week on top of the usual month. My school starts on September 24th, but my visa states permission to enter on August 17th. I've read that you are only allowed a month before the beginning of school, but my visa is saying differently!  Should I assume its wrong and enter on the 24th or is it okay for me to enter on the 17th?   Thanks. How do they decide Blu-ray/DVD release date? I think my question is different from .  Some movies' productions release Blu-ray/DVD so early after theatrical release, others will take time. How do they decide for the Blu-ray/DVD release date? See how many duplicates of a post? Can I see how many duplicates there are of a post? I.e. how many posts have been labelled as duplicate because of this post? If so, how? And can I also get a list of these duplicated posts? Possible Duplicate:                If in a multiple linear regression (enter method) the general model isn't significant (F>.05) but one of the predictors is significant (β&lt;.05), should I consider it as a significant result? I understand the whole concept of Rencontres numbers but I can't understand how to prove this equation  $$D_{n,0}=\left[\frac{n!}{e}\right]$$  where $[\cdot]$ denotes the rounding function (i.e., $[x]$ is the integer nearest to $x$). This equation that I wrote comes from solving the following recursion, but I don't understand how exactly the author calculated this recursion.  $$\begin {align*}  D_{n+2,0} &amp; =(n+1)(D_{n+1,0}+D_{n,0}) \\ D_{0,0} &amp; = 1 \\ D_{1,0} &amp; = 0  \end {align*} $$ Lots of questions in the reopen queue I've always wondered what the deal was with Darth Vader's "crotch door". Does he use a catheter (due to all his injuries) or does he simply flip the door up and whip it out? I've never really seen the the back of his suit under his cape to see if there a corresponding "back door".   It doesn't look like he carries a catheter bag nor a colostomy bag in his suit or anything.     Is this addressed anywhere either in canon or legends sources?