I've read somewhere that every natural $n$ can be expressed using only Fibonacci's numbers which index is superior or equel to $2$. For example $$ 25=F_{8}+F_{4}+F_{2} $$  I wonder if there exists a proof of this statement because I have no clue about it. Maybe an induction ?
Can anyone hint me to prove:  $\forall n\in \mathbb{N}: \exists$ Fibonacci numbers $ F_{i_1},\ldots,F_{i_k}$ such that: $$\sum F_{i_k}=n$$    Note: Every Fibonacci number can appear only once. Thanks
Units and Nilpotents Filtering points in GeoServer layer Delete specific line number(s) from a text file using sed? pass 1 environment variable using sudo Remove the shortcut of an application that was installed using Wine from the Unity Dash How do I remove an application shortcut from the Unity Dash (not from the Unity Launcher)?  My specific issue is I tried to install a game using Wine a while ago. I then uninstalled the game, then uninstalled Wine. Today I found that shortcuts to the game (and other related content that came with it) remain when I push Super and type the game name. (Wine is not visible though.) Global and local variables in R How do you return from 'gf' in Vim I am using Vim for windows installed in Unix mode. Thanks to this site I now use the gf command to go to a file under the cursor.  I'm looking for a command to either:   return to the previous file (similar to Ctrl+T for ctags), or   remap gf to automatically launch the new file in a new window. Inexact measurement and wavefunction collapse As is usually said, measurement of an observable $q$ leads to collapse of wavefunction to an eigenstate of the corresponding operator $\hat q$. That is, now the wavefunction in $q$ representation is $\psi(q)=\delta(q-q_0)$ where $q_0$ is result of measurement.  Now, in reality measurements are never exact, so the wavefunction should not have that sharp peak. Instead it should be broadened depending on precision of measurement. In this case, can we still introduce an operator, an eigenstate of which would the new wavefunction appear? Is it useful in any way? Or does the new wavefunction depend too much on the way it was measured so that each instrument would have its own operator? How would such opeartor look e.g. for a single-slit experiment? How do I put an image texture on a material in Cycles? How can I vertically align the numerators of two fractions? I am writing the following formula:  % arara: lualatex  \documentclass{article} \usepackage{mathtools} \usepackage{unicode-math}  \begin{document} \[y_\mathup{COG}=\frac{h}{3}\frac{l_\mathup{c,a}+2l_\mathup{c,i}}{l_\mathup{c,a}+l_\mathup{c,i}}\] \end{document}   which yields:    I don't like the way, the h and l_\mathup{index} is vertically aligned. Is that typical behavior? Good style? Could I avoid that? Careers: What does "allows remote" mean? When a posting is marked "allows remote", does it mean work in the office some days, work from home some days?  Or does it mean you could actually live out of town and maybe fly in as needed?  The reason I ask is I'm really interested in public facing software rather than internal/enterprise stuff, and most of that seems to happen in a few very expensive cities.  Given it's so hard to hire in places like SV/NYC, it seems like it could be a good scenario for companies if a developer had a really solid list of accomplishments, but I don't know how many companies think this way. Does there exist a matrix $\mathbf{A}\in\mathbb{R}^{3\times3}$ such that $\mathbf{A}^{2}=-\mathbf{I}$? Mark my own question as favorite? Can CBC ciphertext be decrypted if the key is known, but the IV not? Dual boot Win 8 / Ubuntu loads only Win My computer is a Sony Vaio Notebook. I used Ubuntu and Win 8 with dualboot but windows somehow managed to remove the menu to choose the OS at startup. I think the BIOS is UEFI. Secure boot was disabled already and Boot Repair somehow fails.  How can I fix this? A cyclic group is a group that is generated by a single element.  That means that there exists an element $g$, say, such that every other element of the group can be written as a power of $g$.  This element $g$ is the generator of the group.  Is that a correct explanation for what a cyclic group and a generator are? How can we find the generator of a cyclic group and how can we say how many generators should there be? How do I go about porting Ubuntu Touch to different devices? I would like to port Ubuntu Touch to a device that is currently not supported. What are the requirements for doing this and where do I begin? In the experiments for photoelectric emission, the light is incident on one face of the emitting plate, for example the anode, when determining the stopping potential. The electrons are emitted by this face of the plate. Why are the electrons emitted in the direction of the incident light, and not opposite to it?  I have read a little about photons (the key words being, a little), and from what i gather, they have an intrinsic momentum $p=h/\lambda$, which is transferred entirely to the particle that absorbs it (along with the energy $h\nu$). This should imply that the electrons try to move further into the cathode, and not escape it! What is the fault in the reasoning?  Thanks! How do I determine the size of my array in C? Populate an entire pre-configured workspace in i3wm Is it possible with I3Wm to populate an entire workspace with determined size, position and commands for each window?  I know I can do i3-msg "exec xterm -e 'ls; bash'" to create a new window, but I can't specify his position and size.  Maybe there is also an alternative by declaring via JSON the layout or something like that?