For all $n\geq 1$ we have a set $\begin{Bmatrix} 0,1 \end{Bmatrix}^n$  that only consists of $0$'s and $1$'s. Let $G_{n}$ be the set containing all elements except elements where two $1$'s are standing next to each other.    So e.g. $G_{3} = \begin{Bmatrix} (0,0,0),(1,0,0),(0,1,0),(0,0,1),(1,0,1) \end{Bmatrix}$ Now let $a_{n}$ be the amount of elements in $G_{n}$. So $a_{3} = 5$.     I have to prove that $a_{n} = a_{n-1} + a_{n-2}$ for all $n\geq 3$.  I know that each set contains $n-1$ elements ending with $1$ as its given that two $1$'s can't stand next to each other. Now  let $x$ be the elements ending with a $0$. We can define that $a_{n} = n-1+x$. But if I fill this in $a_{n-1}$ and $a_{n-2}$ it seems that I can't get anything out of it.
I am studying for an exam and I am having trouble with this practice question:  In this question, we consider finite bit strings that do not contain $00$. Examples of such bitstrings are $0101010101$ and $11110111$. For any integer $n\geq 2$, let $B_n$ be the number of bitstrings of length $n$ that do not contain $00$.   Determine $B_2$ and $B_3$. Prove that $B_n = B_{n-1} + B_{n-2}$ for each $n\geq 4$. For each $n\geq 2$, express $B_n$ in terms of a Fibonacci number.   Any help is greatly appreciated
Use fixed keyboard for a given app I would like to tell my Android that when I'm in a given app it should switch to a given keyboard. My use case is that I want to use Hacker keyboard with Connectbot. Correlation with a constant I recently printed a paper that asks to prove the "amazing" claim that for all $a_1,a_2,\dots$  $$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$  and thus (probably) that  $$\zeta(3)=\frac{5}{2}\sum_{n=1}^\infty {2n\choose n}^{-1}\frac{(-1)^{n-1}}{n^3}$$  Since the paper gives no information on $a_n$, should it be possible to prove that the relation holds for any "context-reasonable" $a_1$? For example, letting $a_n=1$ gives  $$\sum_{k=1}^\infty\frac{1}{(x+1)^k}=\frac{1}{x}$$  which is true.  The article is Is the sorting algorithm used by .NET's Array.Sort() method a  algorithm? The layman's question. Is there a maximum temperature that can be reached by air on Earth due to global warming? For example, one that would prevent living organisms from functioning? I read somewhere that with the increase of the sun's brightness by 10%, the average temperature on our planet will be 47 C. This, however, only for about 600 million years. Would global warming enable reaching such a high average temperature? If every ideal is principal show the same is true for the image of a homomorphism. Alternative to Adobe Illustrator Can you recommend a good alternative to Adobe Illustrator?   I am learning vector based graphic design, but I am not ready to make such a large investment as to purchase a copy of Adobe Illustrator. Also, Illustrator might be a little more power than I need at the moment. I am using a Macintosh. I understand that the architecture of Convolutional Neural Networks (CNN) and Feed forward (FNN) are quite different. And that CNNs use pooling and filters of shared weights over a patch of the image. I am not so clear on the core convolution operator (1):   If anyone could link me to an explanation, I have looked at the Colah blog and Nielsen's online book, and I understand what it is doing but don't understand the convolution operator.  Also, it looks quite similar to the the core FNN function is there and difference? (2).  (1) convolution operator is $a^1 = \sigma(b + w*a^0)$ which is equivalent to:  $a^1 = \sigma(b + \sum^4_{l=0}\sum^4_{l=0}w_{l,m}a_{j+l,k+m})$  (2) feed forward operation:  $a^1_i = \sigma(\sum^n_{j=1}w_{ij}x_j +b_i)$  Many thanks  Sources: functions taken from: How can I let player click text to use a command? I want players to be able to click text to run an OP command even if that player is not an OP. For example something like:  Console: Do you want to enable Fly? [If you do, Please click here]  *Player1 clicks that text*  Console: /fly Player1   Is something like this possible? What to do with meta questions referring to currently non-existing features? I sometimes come across such questions like that:    The "Not a real question" close reason doesn't exist for long time. It doesn't make sense to answer, comment, or even up/downvote (since there's nothing more to vote for/against it).   Shouldn't such questions be locked as historical ones? Should I flag them for moderator attention or just ignore them? How to find all the subclasses of a class given its name? If I have a point which I want to hit at the end or during a parabolic arc, how would I calculate the needed x and y velocity? What are LiDAR returns? Can someone help with an explanation and some references, in order to understand the nature of return pulses within a LiDAR system. How can I make a script (or make some changes) to allow me to turn the screen off in Ubuntu (12.04) whenever I want, the way thevscreen can be turned off in Cellphones?  And how can I make a combination of keys (a shortcut) to execute that script?  I want to conserve power.  I've seen this question    but there is no hotkey for it.  Moreover, in the answer, it is said that there are many ways to accomplish this so I'd like to know more. Is quantum entanglement affected by time dilation? Is quantum entanglement affected by time dilation? Let's say one of the entangled pair is accelerated to very high speed. When both the entangled particles are observed at the same time, will they have the opposite spin? Can Windows use a Linux shell instead of cmd? How do I vertically center a single cell within a table? Pass extra parameters to an event handler? What is the best way to convert an array to a hash in Ruby In Ruby, given an array in one of the following forms...  [apple, 1, banana, 2] [[apple, 1], [banana, 2]]   ...what is the best way to convert this into a hash in the form of...  {apple =&gt; 1, banana =&gt; 2} Spoiler warning!  During the first escape attempt in Portal 2 (when GLaDOS is still in charge) she opens a door for you which you can choose to go through instead of following Wheatley (the blue-eyed ball). What happens if you go there?