Divisibility Sequence Puns

Disclaimer: I do not know the solution to this problem. It is item (iii) in Richard Stanley's big list of objects counted by Catalan numbers, so I am pretty sure there is a nice solution.

Let A(n) be the number of sequence of the form

1, a(1), a(2), … , a(n), 1

such that for each i in {1, … ,n},

• a(i) is a natural number which is 2 or more
• a(i) divides a(i - 1) + a(i + 1). Here, a(0) refers to the 1 at the beginning of the sequence, and a(n+1) refers to the 1 at the end.

Prove that A(n) is the n^(th) Catalan number. (From the definition, it is not even clear A(n) is finite!)

When n = 2, the valid sequences are 1231 and 1321.

When n = 3, they are 12341, 12531, 132321, 14321, 13521.

Let n, m ≥ 1, and d = gcd(n, m). Then gcd(F(n), F(m)) = F(d).

This is by far the most interesting property of the Fibonacci sequence, and the fact that Matt hasn't mentioned it makes me question his credibility.

https://en.m.wikipedia.org/wiki/Fibonacci_number#Primes_and_divisibility

Here's the proof.

I'm getting hung up on this part

> Because the Fib sequence is infinite and there are only n^2 possible ordered pairs of integers modulo n, two such pairs (F_j,F_j+1) must be congruent: F_i \equiv F_i+m and F_i \equiv F_i+m+1 (mod n) for some i and m.

F_i != i, how can we tell that two of them must be congruent from the fact that there are n^2 possibilities?

Let A = 1/(1 * 2) + 1/(3 * 4) + 1/(5 * 6) + ... + 1/(2007 * 2008)

and B = 1/(1005 * 2008) + 1/(1006 * 2007) + 1/(1007 * 2006) + 1/(1008 * 2005) + 1/(1009 * 2004) + ... + 1/(2008 * 1005)

Show that A/B is not an integer.

I couldn't progress at all , tried using the symmetry of B , tried evaluating etc... Didn't get anywhere.

Help would be appreciated.

What i mean by that is cells in the same tissue are going through different phases at different times but they started from the same cell some time ago so I thought maybe there could be a nice sequence as everything in science seems to have some kind of nice mathematical sequence.

Motivation:

Consider a game in which two shooters take turns shooting at a target. The winner is the one who manages to hit the target first. Say each shot has a relatively small chance of hitting the target. What is the fairest way to take turns?

The Thue Morse sequence is a 0-1 valued sequence that claims to find an answer - this is hard to make precise, but it can be described as “taking turns taking turns taking turns...”.

Using 0 to denote that the first player should shoot and 1 for the second, the sequence is defined by the following iterative process - start with the string 1. At each iteration step, append the binary complement of the string to the end. By binary complement I just mean that you turn all 1’s in the string to 0’s and 0’s to 1’s. Thus the sequence of strings starts as follows:

1, 10, 1001, 10010110, ...

The Thue Morse sequence is the infinite string obtained by iterating this process. The full sequence begins:

01101001100101101001011001101001...

Now I would like to quantify the sense in which this is really the fairest turn taking algorithm possible. To this end I propose a few definitions:

Definitions:

Define, for each 0-1 valued sequence a*n* the advantage sequence Va*n* as follows:

Va*n* := sum (i = 0 to n-1) (-1)^(a*i*+1).

Thus Va*n* simply records at each step n, how many more turns player 1 has had over player 2, or vice versa - with a + sign signifying that player 1 has had more turns and a - sign for player 2.

Define now for each k ≥ 1, the k-th order advantage sequence V^(k)a_n by

V^(1)a*n* := Va*n*;

V^(k)a*n* := Sum (i = 0 to n-1) V^(k-1)a*i*.

Intuitively, the V^(k) can be understood as follows.

• V^(1)a records how many more turns player 1 has had (or vice versa) than player 2 thus far.

• V^(2)a records how many times (with multiplicity) player 1 has had more turns than player 2 thus far.

• V^(3)a records how many times player 1 has had more of a second order advantage than player 2 thus far.

... and so on.

Now, let us define a partial order on the set of 0-1 valued sequences as follows: {a*n} ≥ {bn} if for all n ≥ 0, k ≥ 1, |V^(k)an| ≥ |V^(k)bn*|.

We say {a*n} > {bn} if {an} ≥ {bn*} but not vice versa.

Now we are ready to state the problem. Denote by {T*n*} the Thue-Morse sequence. Is the following true?

Conjecture: {T*n} < {an} for any other 0-1 valued sequence {an} such that a0* = 1. Thus the Thue-Morse se

I switched into Economics for my major recently and I was wondering if any upperclassmen Economics majors could help me figure out whether I’d be screwed over in doing my calculus sequence outside of UT? I want to build a strong base for calculus since I hear it’s important for the upper division courses but I was also considering getting ahead/catching up by completing my calculus sequence elsewhere but I’m not sure if that would hurt me or not? I’d appreciate any advice on this since registration is coming up so thank you!! (Also, if you recommend me taking Calc at UT, how difficult would you say it is? Because that’s something else I was a bit worried about.)

Hello all, I'm just getting started putting together a Eurorack moduar setup. I plan on all the VCOs, VCAs, LFOs, filters & etc. being homebrew, but I'd like to start out with a simple but solid sequencer just to have something to get all the goodies moving. To that end, I have my eye on the Division 6 Dual Mini Sequencer. I was a professional vacuum tube guitar amplifier builder for over a decade, am currently a web developer, and have done a fair bit of microcontroller programing and general electronic tinkering, so I'm pretty confident I can handle the technical aspects of assembly - just want to know if anyone has had experience with this unit and if it seems like a solid choice for $90. If you know a killer alternative to suggest, that's cools too. Thanks!

Is there a name for a sequence of divisions, subtractions, and/or multiplications - like a "series" is a name for a sequence of additions?

Other than sum, product, quotient, difference?
Would those even be applicable?

I've been meaning to learn Conway's Doomsday algorithm for the longest time now, mostly because it's just fun to be able to do this, but also because I thought it'd make a neat party trick with which I could demonstrate the usefulness of mathematics. Are there any others one can learn that don't require extreme mental arithmetic faculties? Obviously multiplying/factoring large numbers in one's head is always neat, but I don't think it's something accessible to an average person (without serious amounts of effort and training).

Natural numbers with multiplication are a partially ordered monoid via the divisibility relation in a natural way, with 1 being the least and 0, which is divisible by anything, the greatest element. This partial order is also a complete lattice, with gcd and lcm as finite infima and suprema, and 0 the supremum of any infinite subset. In particular, with this ordering we have lim_n a_n = 0 for any strictly increasing sequence, and ΠA = 0 for any infinite set A.

The lattice of ideals of Z is the mirror image of this, so any way of saying "(0) is the smallest ideal" is basically saying "0 is the largest integer (in this particular sense)", but I was idly wondering if there are more interesting or surprising contexts where you get behavior that looks kind of like e.g. lim_n p^n = 0 or Π_p p = 0? Statements like "if P is true mod n for infinitely many n, then P is true mod 0" should also qualify, but I don't know if there are interesting examples with this generality, instead of the weaker versions with "... true mod p for every prime p" as the condition.

Hi everyone. I was the guy that deduced and popularized what has recently become known as the “Roman master plan theory” on this subreddit, a theory that Ty and Daniel recently confirmed is correct. In the Alt-Shift X podcast where they confirmed it, they said that they felt they weren’t being subtle, and yet a ton of people seem to have completely missed this central part of the plot of Leviathan Falls. I think a major part of why I understood what the authors were going for is because I understood The Dreamer chapters in the book, and therefore understood the alien biology that they were trying to explain. A biology that is so alien that I think it actually challenges a lot of what we consider as “life” in the first place.

So, I thought that I would make a thorough post on the evolution and history of the Gatebuilders, so that other fans can understand their nature and the overall alien plot line of the Expanse. My background is in biology and neurology, so I think I might be decently suited to make a post like this. I won’t go into detail about their “master plan” - there are multiple recent threads on that. Instead, this will delve into their biology and evolutionary history. I will refer to them as the Gatebuilders throughout this post because that is the name we seem to have collectively decided on as a fan base, but by the end of this post, you’ll see why I think that instead of calling them “Gatebuilders”, we should instead call their species “Leviathans” - a name that I think suits them far better, both based on their origin and because they did far more than just build the gates, which probably wasn't even initially deliberate in the first place.

This will be long. VERY long. I apologize for the length, but there was a lot to discuss here. There’s a tl,dr below.

I will also say, for clarification, that I repeatedly refer to them as “Gatebuilders”. But as Elvi points out, the plurality is a misnomer. The word “hive mind” is meaningless - all there is, is mind. Just as you and I are not a “hive mind” of the billions of neurons that contribute to our consciousness, so too were they not a “hive mind” of the myriad nodes that contributed to theirs. They were a mind, and therefore “Gatebuilder” or “Leviathan”, singular, would be a more accurate term. But because that is almost grating to our human perception, I’ll continue to refer to them in the plural.

Most of this information is contained within The Dreamer chapters and Elvi’s followup, expl

This is the sixth part of my write-up. You can read the other parts here.

• Part 1 - Beta and Vanilla

• Part 2 - Burning Crusade

• Part 3 - Wrath of the Lich King

• Part 4 - Cataclysm

• Part 5 - Mists of Pandaria

#Part 6 – Warlords of Draenor

The Warcraft Movie

This might seem like a bizarre topic to start with, but stay with me here. It all links together.

On 9th May 2006, a Blizzard press release announced the production of a live-action movie set in the Warcraft universe, in partnership with Legendary Pictures. Fans were euphoric. Blizzard’s cinematic trailers had some of the best CGI in the world. Even today, they have never released a bad one. Fans wanted something like that, only 90 minutes long.

>"We searched for a very long time to find the right studio for developing a movie based on one of our game universes," said Paul Sams, chief operating officer of Blizzard Entertainment. "Many companies approached us in the past, but it wasn't until we met with Legendary Pictures that we felt we'd found the perfect partner. They clearly share our high standards for creative development, and because they understand the vision that we've always strived for with our Warcraft games, we feel there isn't a better studio out there for bringing the Warcraft story to film."

However good their intentions may have been, the film would linger in production hell for a decade before seeing the light of day. It was scheduled to hit theatres in 2009 under the direction of Sam Raimi (of Spiderman fame), but it was still only in its early stages when Blizzcon 2011 came around..

Uwe Boll, grim reaper of video game adaptations, tried to get h

My first powered DIY module. It uses an Arduino Nano.

It's a clock plus 2/4/8 clock division.

Hand drilled out of alignment holes.

Too many holes.

Atrocious wiring nest.

I'm still proud of it :)

https://preview.redd.it/z93q0zxd1de81.jpg?width=3024&format=pjpg&auto=webp&s=f07076783a9d1fb0c04e7beca485cf20980c618f

https://preview.redd.it/f9qi07yd1de81.jpg?width=3024&format=pjpg&auto=webp&s=bbe50cdb9389cc533feee3a507e747aa9e761214

*Warning this is very long!

Hello everyone! Your girl is back for 2022 and I will be trying to keep a more consistent schedule this year. To kick off 2022 I figured we could take a closer look at one of UFC 270's headliners, interim champ Ciryl Gane. I hope you all enjoy, have a wonderful night of fights!

Introduction:

Interim UFC heavyweight champion Ciryl Gane will look to unify the belts against undisputed champion Francis Ngannou on Saturday night. Ciryl Gane's ascension through heavyweight MMA has been blisteringly fast, a mere 10 fights into his professional career he wears UFC gold. It has been marked by dominant decisions, fascinating finishes, and skills rarely seen at heavyweight. Heavyweight MMA is notoriously underdeveloped relative to the smaller weight classes, Ciryl Gane does not quite bridge that gap but he brings the weightclass much closer to the rest. With that said, let's take a look at how he does it.

Footwork:

Much of the hype around Ciryl Gane has been around his footwork, about how "agile" he is despite his size. Ciryl Gane is what is known as a "ring general" at 265. What this means is that he controls where the fight takes place and has excellent awareness of where he is in the cage.

https://gfycat.com/terrificdeepgopher
In this clip you can see that as Lewis begins to pressure Gane, Gane takes 2 steps backwards and circles around, retaking the center of the octagon. He reestablishes himself with a heavy low kick and the fight continues with Gane in control.

Taking yourself off the center line is one of the most important things that a fighter can do. When a fighter is moving straight backwards, their opponent has a direct line to them. They will also end up running into the cage very quickly which immediately removes half of their movement options. Striking is often about angles and if you can force your opponent to an angle where they cannot line their shots up with your head that is usually a good idea.

Gane is one of the best heavyweights when it comes to controlling where a fight takes place. In the previous clip we saw Lewis attempt to dictate that the fight would take place with himself in the center with Gane along the cage. Gane circled out to avoid that because he wants to fight from the center with Lewis along the cage. With this idea in mind lets take a look at the following clip:

[https://gfycat.com/educatedweightyjerboa](https://gfycat.com/edu

Artist: Kanye West | Album: Donda

Alternate cover art he has tweeted:

Alternate cover 1

Alternate cover 2

Listen: SPOTIFY | APPLE MUSIC | TIDAL | YOUTUBE

Background by /u/Kitchen_Ur_Lies

The culturally ubiquitous Kanye West, now legally just “Ye”, seemed to be working on everything besides music after releasing his latest solo effort, ‘Jesus Is King.’ A post-Jesus spin-cycle rendition of his cancelled album ‘Yandhi’, it gave listeners what was to be the first of his exclusively non-secular gospel releases “from here on out.” After the uniquely mixed reception to a West effort, Kanye embarked on many ventures outside of music, including helming a creative director role at Gap and casually running for president of the United States. As Kanye’s Gap and very successful Yeezy ventures seemed to make him a billionaire by April 2020, music appeared to be an afterthought for the next iteration of his artistic catalogue.

During an interview with Michèle Lamy in May 2020, cinematographer Arthur Jafa, whose work includes Jay-Z’s 4:44 music video, announced he was [collaborating with West on a single off his previously unannounced record, *God’

Hi there!

Consider the following sequence:

U_{n+1} = U_n / 3 - 2

We could express the sum of terms of U_n as

S_r = sum(U_k for k from 0 to r-1 inclusive) = (3/2 U_0 + 9/2)(1 - (1/3)^r) - 3r

I was wondering if it is possible to find an expression for this sum if instead we had:

U_{n+1} = U_n // 3 - 2

Where "//" represents the result of the euclidian division (i.e. floor(U_n/3))?

The thread itself: https://twitter.com/What46HasDone/status/1484311526580584451

(thread reader version: https://threadreaderapp.com/thread/1484311526580584451)

Edit: Website version (this is where new additions will be added): https://whatbidenhasdone.wordpress.com/2022/01/20/year-one-what-biden-has-done-mega-thread/

I wanted to share its contents here because it's significantly more impressive to see all of the action taken by this current administration (even with a 50/50 senate majority and a lack of a filibuster), as well as how many just good policies have been enacted even if they've not been easily advertised or reported.

YEAR ONE WHAT BIDEN HAS DONE MEGA THREAD

•1.9T American Rescue Plan

•$1400 stimulus checks for adults, children, and adult dependents

•1 year child tax credit expansion - $3600 0-5, $3000 6-17, removed income reqs and made fully refundable

•One year EITC expansion

•$350 billion state and local aid

•$130 billion for schools for safe reopening

•$40 billion for higher ed, half of which must go to student aid

•Extended $300 supplemental UI through September 2021

•Expanded eligibility for extended UI to cover new categories

•Made $10,200 in UI from 2020 tax free

•$1B for Head Start

•$24B Childcare stabilization fund

•$15B in low-income childcare grants

•One Year Child and Dependent Care credit expansion

•$46.5B in housing assistance, inc:

•$21.5B rental assistance

•$10B homeowner relief

•$5B for Sec 8 vouchers

•$5B to fight homelessness

•$5B for utilities assistance

•Extended Eviction moratorium through Aug 2021 (SC struck down)

•2 year ACA tax credit expansion and ending of subsidy cliff – expanded coverage to millions and cut costs for millions more

•100% COBRA subsidy through Sept 30th, 2021

•6 month special enrollment period from Feb-Aug 2021

•Required insurers to cover PrEP, an HIV prevention drug, including all clinical visits relating to it

• Extended open enrollment from 45 to 76 days

•New year round special enrollment period for low income enrollees

•Restored Navigator program to assist with ACA sign up

•Removed separate billing requirement for ACA abortion coverage

•Eliminated regulation that allows states to privatize their exchanges

•Eliminated all Medicaid work requirements

•Permanently removed restriction on access to abortion pills by mail

•Signed the Accelerating Access to Critical Therapies for ALS Act to fund increased ALS research and expedite access to experimental treatments

•Re

(I thought it would be fair, informative and engaging to share Ingram's response here as a top-level post, considering that the original critical review gained significant attention. Text continues in comment section.)

DM48: I’ve been doing a lot of re-evaluation of Ingram's ideas and works and how they may be impacting people's practice. I've researched through enough Suttas myself, and, I believe, being an "accomplished" enough practitioner of the Noble Eightfold Path and Four Noble Truths, I feel comfortable enough pointing out some positives while also fleshing out critiques of the book.DMI: I would suggest re-reading MCTB2 again, as clearly you missed much about it or didn’t remember it (or barely read it) which is understandable, as it is long and complicated. It probably takes a few reads to get a sense of how each section contributes to the others. I will help you out by pointing out the more glaring things you either missed, didn’t remember, or didn’t understand. I will also think about how MCTB2 contributed to any misunderstandings besides being really, really long. Speaking of really long, those familiar with my point-by-point style will have expected this very long reply, and hopefully it will not disappoint.

DM48: This has direct implications for practice, especially people following a Therevada-inspired Buddhist path. Although I think there are some relevant points here for any kind of contemplative.DMI: Worth knowing that my inspirations are quite wide, and, while, yes, clearly in some ways “Theravada-inspired”, in others aren’t, as noted numerous times in MCTB2, including in the first few pages.

DM48: **The positives:**Firstly, I think the positives are that Ingram's book Parts I and II are great.DMI: Ok, thanks. Wish you had remembered them and understood their implications for later Parts, as I will point out below many times, but will take the honest complement.

DM48: They elucidate the core teachings in a very open carefree way that gets people seeing that the path is simultaneously a very serious thing and fun thing. Being moral is happy. Having a unified mind is happy. Being wise is happy.DMI: Ok, those three lines are one of the more trite and superficial summaries of those parts I have seen, and I have seen some bad ones. One of the

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

It is a bit of a common thing in fan discussions to make the good guys and/or the protagonist and protagonist adjacent group of characters to be worse than they actually are by exaggerating things and leaving out context all the while making the villain(s) blameless. It often feels like a way feel better about oneself if someone sees too much of themselves in the supposed bad guys or antagonist. But make no mistake, although I do not actually think all of Soul Society from bleach is terrible and it could honestly be worse, these are plain (or mostly plain) facts.

One of the first things we hear about Soul Society is that it is a paradise. Well, that's a fucking lie. And it turns out that Soul Society has a habit of covering things up and verging on being outright authoritarian, but that is getting ahead of myself. There is an extreme disparity with living conditions, and when you die and arrive at Soul Society, it is a literal luck of the draw as to where you go.

Some of the rougher areas are incredibly poor with children having to steal to get by. And in the worst areas, life is ridden with constant death and violence. In fact, one character had to abandon her sister who was literally a baby since she would not be able to survive otherwise. She would spend a few years trying to look for her sister, but it turns out there may not be a registry of any kind! One may think it is because they do not have the technology, but nope! Soul Society actually does have advanced technology. It just so happens that that technology is mostly concentrated where the nobles and Soul Reapers live.

It may not be the only way to improve life, but it is said that becoming a Soul Reaper/Shinigami is a common way of escaping a life of misery. Except not really since you can still get killed for an arbitrary reason and get away scot free if you have a good enough social standing (and maybe I am reading too much into it, but is apt how this was a case of domestic abuse with a gut like Tosen having pleas of justice ignored, and as revealed in a novel, framed for something and beaten up).

Lots of murder and exploration is another thing that just sort of happens. It may have lessened a bit over the years, but it was an accepted thing to kill citizens under the guise of control (even if it must be acknowledged that those areas are dangerous), and under the l

The division 6 is a very inexpensive sequencer with some limitations. One often noted limitation is that it doesn't have a reset input. I can confirm that feeding the clock and reset from a Mutable Instruments Grids to the clock and xp in of the Division 6 results in a perfectly synchronised sequence. I don't know if this works with other sequencers in general but its value as an expansion for the Grids is a no brainer. You can even use the accent output as an additional percussion trigger if you wish.

(The embargo lifts tomorrow, on the 16th, but nevertheless some people have posted their impressions today already. But expect a lot more in a day.)

DO NOT POST SPOILERS IN THIS THREAD. Only social media impressions by the press.

To start with, the one from this thread: https://www.reddit.com/r/matrix/comments/rgpi1u/first\_reaction\_to\_resurrections\_i\_just\_saw\_on/ that has since been taken down for breaking the embargo:

https://preview.redd.it/o93pkwff7o581.png?width=811&format=png&auto=webp&s=8c9ac51f6cd41accd55958bbe8bc294d355d1ef6

https://twitter.com/headgeek666/status/1470979936638980098

>Dear friend saw #MatrixResurrections tonight and fucking loved every second! Can not wait

https://twitter.com/awalrusdarkly/status/1470974832816832518

>THE MATRIX RESURRECTIONS is Lana Wachowski's bold attempt to reckon with her own cinematic legacy, and she mostly pulls it off. It's a fascinating, ambitious, sometimes frustrating but always compelling thing, and I think I have to watch it a few more times to really digest it.

https://twitter.com/tracesauveur/status/1470969464539783175

>THE MATRIX RESURRECTIONS: Generally positive on this, though not without some reservations. Not sure the things I like about it really come together as a cohesive whole. But still miles better than a lot of other bullshit franchise resurgences.
>
>I should note I will probably watch this again immediately once it’s on HBO to see if it goes down a little easier.

Prediction: This will be a film worth a few rewatches to get the most out of it.

NEW:

https://twitter.com/emilyvdw/status/1471668975129948170

>There was a point in the middle of The Matrix Resurrections where I briefly thought it was the best movie ever made, and, like, I haven’t convinced myself it’s NOT? I lovvvvvvved it. A lotta people are gonna haaaaaaaate. My favorite kind of movie!!

https://twitter.com/davidlsims/status/1471671375698051077

>THE MATRIX RESURRECTIONS is very good :)
>
>**it made me very