A Number Sequence with Everything - Numberphile

[@numberphile]

Check out Jane Street's sidewalk sequence at: www.janestreet.com/numberphile2022 Visit the OEIS at: oeis.org/

[@FebruaryHas30Days]

First reply I use OEIS

[@paulthompson9668]

4:53 The envelope reminds me of the Fibonacci numbers, which has a cosine in it.

[@Ethan-lu7gd]

OEIS is one of my favourite websites, It's always a joy to see videos on the myriads of wonderful sequences it contains! Thank you!

[@maitland1007]

The Jane St thing sounds to me like "Hey, if you are smart and like math, come help us make rich people even richer". Am I wrong?

[@paulthompson9668]

@@maitland1007 It sounds like a cult.

[@rozhenko]

Honored to be mentioned in this video by the great Neil Sloane! Thank you Neil and thank you Numberphile for posting the video.

[@iamthecondor]

To be fair, you've earned it 😅

[@danielg9275]

Awesome when a celebrity reacts to the video!

[@staizer]

What is this sequence like in binary?

[@jonaslarsson5279]

@@staizer It's not based on the digits but on the numbers. I.e. when 10 shows up you don't view it as a one and a zero, but as a ten. Interesting question nonetheless, were you to interpret a 10 as a one and a zero.

[@archivist17]

Thanks for a creative and beautiful sequence, Joseph!

[@nicksamek12]

A beautiful message to end the video with. A lot of math isn't in the destination, but the understanding you develop on the journey.

[@lonestarr1490]

So you gonna tell me, maybe the real math is the friends we made along the way?

[@quintrankid8045]

Shouldn't we generalize that?

[@jomolisious]

Journey before Destination.

[@JorgetePanete]

A 2000 theorems journey starts with 1 statement

[@angelodc1652]

@@lonestarr1490 I was about to say something similar

[@Drej9]

Neil Sloane is an international treasure. With every video he appears in, the content becomes so interesting and engaging. More Neil!

[@Triantalex]

??

[@julesmcbride2692]

"We have the variations, but we don't know what the theme is." What a stellar analogy for mathematical puzzles.

[@aceman0000099]

The music was like someone getting chased, and stumbling, but every time they stumble they manage to run a bit further and the suspense builds

[@vigilantcosmicpenguin8721]

@@aceman0000099 It's a neat effect how the tempo doesn't change, yet it feels like something is getting away from you.

[@matthewdodd1262]

Strangley, even the fun maths is super important. When people find new and weird ways of doing something silly and fun with stuff like this, it can bring forward new ideas which can be used to solve more important problems in mats

[@valdezunderrune394]

Boss: How’s your assignment going? It’s due later today. Me: 0:26

[@AruChanWZ]

This man loves what he's doing. He looks so satisfied at the end of the video )

[@Axacqk]

On a meta-level, it is not that surprising that a sequence defined recursively in terms of _all_ its previous values exhibits interesting behavior. No information is ever lost - every element of the sequence will be used infinitely often in computing subsequent elements. The sequence just meditates upon itself forever, without ever losing any "insight" once gained.

[@DiamondzFinder_]

I was literally just rewatching the planing sequence video when I got this notification.... This guy is so satisfying to listen to, and the sequences he shows us are so fun! Love it

[@DekarNL]

Totally agree. Would love to see progress made into understanding these types of sequences.

[@maynardtrendle820]

Look up the 'Experimental Mathematics' KZbin channel, and you'll find some Zoom lectures from Neil regarding all kinds of OEIS sequences. Also, a lot of other cool videos! It's a small channel from Rutgers University, but Neil is a constant on it.

[@Triantalex]

??

[@DiamondzFinder_]

Thanks for the recommendation! @@maynardtrendle820

[@JaxonHaze]

I love this guy he has the most calming voice

[@fleabag500]

neil's videos are some of my absolute favourites. he has an amazing, relaxing voice.

[@DekarNL]

Love Neil and the OEIS. Used it for a math puzzle the other day :)

[@MushookieMan]

That's cheating

[@teemuaho4807]

I often think about math instead of actually concentrating on whatever lesson is at hand and whenever i figure out a cool sequence or constant i plug it in the OEIS to see if there's any cool formulae or connections with other numbers

[@Triantalex]

??

[@dit-zy]

Neil is so excitedly passionate and I just absolutely love it! He's adorable and so interesting to hear from 💕

[@kikivoorburg]

Neil is awesome, his excitement is super contagious!

[@gandolph999]

Your enthusiasm and fascination with this Inventory Sequence are pleasantly infectious. It is interesting.

[@2Cerealbox]

Two great quotes from this video. "Here, we have the variations. But we don't know the theme." "Maybe in itself its just a sequence. But who knows where it will lead."

[@thomaschevrierlaliberte5884]

Those rows of book on the shelf facing him seem like such a lifetime of mathematical passion.

[@txikitofandango]

It's never a bad time to thank Neil Sloane for his contributions which have helped mathematicians around the world for generations.

[@Xamimus]

Neil Sloane is one of the best Numberphile presenters!

[@Bethos1247-Arne]

Every video with this guy is a must-watch.

[@applechocolate4U]

This is without a doubt my favorite numberphile video

[@randy7894]

Neil is a math poet. I love his video's.

[@AngeloEduardo-gs6yv]

Kkkk😊

[@derekhasabrain]

I show up to every video with Neil Sloane and I always will!

[@SpooNFoy]

The worst Neil Sloane video I've ever watched was excellent. Can never have too much of this man.

[@thehearth8773]

I can't help but notice, there's also the little digits Neil draws to say which number each term refers to. I wonder how the sequence would change if you included those! It'd be kind of like the look-and-say sequence, but without grouping the numbers.

[@altejoh]

I'd be really curious to see a Fourier Transform of this series, it reminds me a lot of energy levels and spectra from chemistry/physics.

[@aceman0000099]

I don't know if it's possible

[@robertr7923]

Me too! Should be doable in a program. You can find the sequence on the OEIS

[@bur2000]

@@aceman0000099 You'd have to interpolate the original sequence to get a continuous function, I think. Fourier transformation of discreet values doesn't make sense - unless I'm mistaken.

[@marclink0]

@@bur2000 as far as I know, both Discrete Fourier Transform and Continuous Fourier Transform exist

[@AbandonedMines11]

This was all so very fascinating. I’m a pianist, too, and found the musical tie-in to be very intriguing.

[@j.thomas1420]

Boulez would certainly have liked to make something from this. The closest piece for piano I know to that sequence is Ligeti, Devil Staircase.

[@marvinabarquez8915]

I see you went down the YT alg rabbit hole too

[@jhoylangoncalves3127]

I just love this gentleman, his passion about numbers and sequences are just intoxicated

[@LMacNeill]

I could listen to him talk for hours. Always interesting and engaging -- I've watched every video you've made with him. I do hope you'll have more videos with him in the future.

[@ComboClass]

The OEIS is an amazing resource. One of the best websites in existence

[@legendgames128]

Eyyy! Combo Class spotted!

[@mathphysicsnerd]

Always love to see a Sloane video, the man makes my day

[@jimmyh2137]

I would love to look at the same sequence with a variation where you also count the "index". So it would go: 0_0 (zero "zeroes") 2_0; (two "zeroes" because you got the "index") 0_1; 4_0; 1_1; 1_2; 0_3; 6_0; 4_1; 2_2; 1_3; 2_4; 0_5; 8_0; 6_1; 5_2; 2_3; 3_4; 2_5; 2_6; 0_7; ... First entry is always 2n (you always have one index for 0 and the last entry) but the pattern for other digits looks very different, or maybe we can find some connection with the "base" sequence!

[@SgtSupaman]

I thought at first that is how the pattern would work in the video, since he wrote those subscripts and asked how many we could see, but apparently, they were just there to help him explain/keep track of the meaning of each digit. The sequence in the video could be written without the subscripts entirely (and in one continuous line). An interesting aspect of doing it in a way that includes the index is that you are guaranteed that the numbers in the columns will always increase by at least one for every additional row, because the index is will always be present in each row. By the way, slight error in your index-counting sequence. The 4th line should have "2_4;" instead of "1_4;" (there is a 4 in line three and a 4 earlier in line four), which would change your 5th line to 8_0; 6_1; 5_2; 2_3; 3_4; 2_5; 2_6; 0_7; So far, this suggests each row will stop (by hitting a 0) at 2n-1.

[@jimmyh2137]

@@SgtSupaman Oh yeah, fixed now.

[@mairsilpretner6119]

Neil is always an amazing guest, his love for these sequences is very infectuous

[@shanehebert396]

Everybody needs someone who talks about them like Dr. Sloane talks about sequences.

[@davidmurvai40]

The content is amazing but his speaking voice is absolutely wonderful ❤. So soothing and such a captivating style.

[@I_Was_Named_This_Way...]

I made something for this in Excel, took about an hour to make but it works flawlessly

[@TranscendentBen]

8:54 He mentions John Conway - it was just after the first minute that I thought of the look-and-say sequence that Conway had analyzed and apparently made famous. My goodness I should have been a mathematician! I could sit around, drink coffee and come up with sequences like this all day! ;-)

[@EvilSandwich]

After I listen to this absolutely fascinating discussion, I have come to the conclusion that, for humanity, mathematicians are quite possibly one of the most important and vital community of completely batshit crazy people in the world.

[@thebrewster]

"it's very irregular, and wonderful" love the enthusiasm, new to this channel.

[@simonblake1434]

Love a Neil Sloane video - thank you Numberphile :)

[@inrlyehheisdreaming]

Regardless of the inherent value of the sequences themselves, the best of these videos is seeing how happy they make him!

[@Reggiamoto]

Videos with Neil Sloane are always a highlight. One question I have is whether every number will appear? Isn't it possible that one number gets skipped by all previous numbers, so you'd always have to take inventory for the same number from that point?

[@christianellegaard7120]

No, I don't think so. The zeros take care of that. Every time you take inventory there is one more zero. So all the numbers appear in the first column.

[@mellowyellow7523]

rewatch around 2:30 he says the next line will always be the next number

[@Boink97]

Apart from the trivial appearance (when the numbers appear because of the zeros) - do we know if every numbers appears at least once more?

[@jimmyh2137]

@@Boink97 that's a great question, we need answers!

[@SgtSupaman]

@@Boink97 , due to the fact that numbers are constantly being added and never taken away, this doesn't seem as though it would ever skip any number infinitely, even if you don't count the number's required initial appearance. We can see that the amount of each number (the columns formed in the way he lays it out) will continue to increase. They may not increase on every row, but they all increase. So, once a number gets a 1 in its column (which it has to, given the "trivial appearance"), it will certainly increase from there.

[@тими]

The plot looks like a banger 808 sample 👀 Need to check it asap!

[@connorohiggins8000]

I really enjoy the OEIS videos. I got a sequence accepted a few years ago (A328225) after one of these videos. This just reminded me that I never figured out why my sequence looked the way it did when it was plotted. I would love to hear some thoughts. I am not a mathematician in any form, so it could be absolutely nothing.

[@dallangoldblatt7368]

I'm gonna look, I'll get back to you in a bit

[@LunizIsGlacey]

Oh wow, that's quite cool! Seems like such a strange rule, but the plot is very interesting!

[@connorohiggins8000]

@@dallangoldblatt7368 Thanks Dallan

[@kindlin]

@@connorohiggins8000 What does prime(n) mean? Checking to see if it's prime? Does it return 1 or 0? But then, what would prime(prime(n)) be? How does that sequence work? (This is just a formula question, I simply do not know what prime(n) might return.)

[@connorohiggins8000]

@@kindlin Hi, so prime(n) means the nth prime, prime(1) = 2, prime(2) = 3, prime(3) = 5 .... If n = 2 then prime(prime(n)) = prime(3) = 5. It is a bit of a weird sequence.

[@Hamuel]

I adore seeing Neil explain more sequences!

[@Mechanikatt]

Oh boy, more Neil!

[@thewatermelon3831]

I'm new to this, and i have a few questions if anyone may be so kind to answer: 1. What is the point of the sequence? 2. Why use a marker over paper? 3. What was so extraordinary about the music?

[@Pfhorrest]

Even before the big obvious leap in the curve that you called attention to, I was already noticing a smaller leap in the earlier part of the curve, and now looking at the larger curve with the big obvious leaps in it there are even more clearly a series of ever-smaller leaps near the beginning of the sequence too.

[@SuperYoonHo]

I love vids with Neil Sloane!!!😍

[@YG-ub4dk]

Always love the Neil Sloane sequences videos :)

[@mrwizardalien]

I didn't know you could download those as MIDI! I immediately went off to go make some sequence music!

[@reidflemingworldstoughestm1394]

Love the Sloane videos.

[@hindigente]

It's impossible not to chuckle at ~5:00 when Sloane shows the sequence's unexpected behaviour.

[@andybaldman]

Why?

[@rayscotchcoulton]

I love his reply to Brady's comment at that point when he says it's irregular... and wonderful. The way he says that makes me smile.

[@hindigente]

@@andybaldman Because of both how unpredictable the sequence's envelope turns out to be and how endearingly Neil Sloane presents it.

[@vigilantcosmicpenguin8721]

Just when you thought things were making sense.

[@FloydMaxwell]

Great background music for a suspense scene

[@builder1013]

You can also just take any number and “take inventory” with the digits you already have and going from there, possibly even summing up the digits of each inventory count to make for an interesting game.

[@builder1013]

Like, for instance, a section of the Fibonacci sequence, the letters of a word, digits of pi, other sequences, or just random numbers to see what you get.

[@builder1013]

You could also try taking inventory of only the digits in the last count and see what happens. I had a number that looped back around after 16 counts.

[@builder1013]

Also if there are duplicate replies, that’s my bad, the Internet isn’t the best here

[@builder1013]

In fact, if you start with 13120 the first inventory count will be 13120. (one 0, three 1s, one 2, two 3s, and no 4s).

[@questioneverything55]

his chuckle is Epic

[@guillaumelagueyte1019]

After seeing the underlying mathematics of the look-and-say sequence, I most certainly hope we will be able to find and explain since structure with this one as well. What an absolute beauty

[@MichaelGrantPhD]

If I were a greedy inventory taker, I wouldn't re-start my inventory when I get a zero. Instead, I would immediately jump to the number corresponding to the count I just arrived at. For example, if I'm currently counting the number of 8's, and I count 3 of them, I would count the number of 3's next. Of course I know that will be one more than the last time I counted it. So I never really have to re-count anything, I'm just incrementing by one every time.

[@zipzorp-eh1ey]

I really didn't understand, could you give an example of how it would change the sequence, please?

[@legendgames128]

So jump to the count you last had. 0_0 1_0, 1_1 2_1, 1_2 3_1, 1_3 4_1, 1_4 Hmm... being greedy from the very beginning results in a less interesting sequence over all.

[@hosz5499]

A Great game for elementary students, to build concepts of sequence, logic, infinity, graph, etc etc!! I will do this in my next math lecture

[@senthilkumaran5255]

Is this somehow connected to the Mandelbrot set? That's what struck me when I saw "this sequence has everything" and the fundamental unpredictable yet beautiful nature of it seems very similar to Mandelbrot. The fact that when converted to music, it seems to follow a pattern of highs to lows with slight variatons for each block/chunk is like penrose/fractal tiling that repeats infinitely with small variations, aperiodic yet beautiful!

[@Eagle0600]

That question at the end, and Neil Sloane's response, highlights an important point; mathematics like this is exploration. By its nature, you don't know what you'll find when you're exploring until after you've found it. So whether or not you're exploring in search of beauty, or for fun, or for something of some other value, you can't really place a value on the exploration itself.

[@joaobaptista5307]

You could say, in some cases, that exploration is an end to itself.

[@SgtSupaman]

Even just hearing this guy say "Here's what we have so far... blank paper" with that smile is enough to interest me.

[@yami_the_witch]

I feel like this sequence could be great for encoding, the arbitrary erratic nature of the sequence is one part. But also, there is no concrete way to skip to a specific result with an input of n. You *have* to compute all of the previous terms to get your term. As n get's larger and larger, it's going to get more and more difficult to brute force the encoded sequence.

[@ishangoel5794]

you mean hashing?

[@mikeness5074]

This guy is really the OG of calculation!!!!

[@MisakiiG]

His voice is fing magnificent

[@thelocalsage]

i got very excited about this and was playing with it, started one where i did inventory but inventoried numbers greater than or equal to the index (later found it in OEIS already) but i found some fun patterns and would love to know why they’re like that! there was a fractal pattern that emerged and also there was another OEIS sequence correlated with the peaks. would love to hear someone like Neil explain why

[@devjock]

The sequence looking for a killer app. Quite distinctly put, Mr Sloane!

[@JacobCanote]

The patterns are beautiful.

[@davidvegabravo1579]

I know nothing about math, but i love this guy!

I wonder how it changes in different base numbers

[@softy8088]

I immediately thought that the "Stock" in each chunk might be incomplete if you hit a zero count for a number n, but n+1 appears previously. It turns out this happens! At a(46) we take stock of 8's, and there are zero, so a(46)=0, and we start a new chunk. However, a(39)=9, so we stopped taking stock too early and we aren't counting the 9's and maybe we should be. So my alternate algorithm isn't to reset the chunk when we hit zero, but reset after looking for the number one more than the maximum in sequence so far. (This is guaranteed to be zero of course.) I did this in Excel. The sequence you get from this is identical up to a(46)=0, but the original sequence has a(47)=8 (a new chunk started), but my sequence has a'(47)=1 (we count the 9's, then the 10's, then start a new chunk). Well, is this sequence any more interesting? Turns out, not so much. Zeros take over pretty quickly because the maximum grows fast but leaves a lot of missing numbers.

[@dewaard3301]

The way Neil eases us into his sequences makes me certain he's got grandkids that he loves to read to.

[@veggiet2009]

I'm immediately interested in adding more rules and see what happens, like what happens if you begin with a random seed number? Like start with 3: 3, 0 1, 1, 0, 2, 2, 2, 1, 0 3, 3, 3, 4, 1, 0 4, 4, 3, 4, 4, 0 5, 4, 3, 6, 6, 1, 2, 0 6, 5, 4, 6, 6, 2, 5, 0 7, 5, 5, 6, 8, 5, 6, 1, 1, 0

[@veggiet2009]

You could also add and extra question before 0, like "how many primes are there?" 0, 0 0, 3, 0 1, 4, 1, 0 1, 5, 3, 0 3, 6, 3, 0 5, 7, 3, 0 8, 8, 3, 0 9, 9, 3, 0 10, 10, 3, 0 11, 11, 3, 0 13, 12, 3, 0 Well that didn't do what I expected

[@veggiet2009]

Going to try adding an additional question after the first 0, before resetting. 0, 0 2, 0, 1, 0, 0 5, 1, 1, 0, 0 6, 3, 1, 1, 0, 1, 1, 0, 0 10, 7, 1, 1, 0, 1, 1, 1, 0, 0 13, 12, 1, 1, 0, 1, 1, 1, 0, 0 16, 17, 1, 1, 0, 1, 1, 1, 0, 0 19, 22, ...

[@bgtyhnmju7]

Neil Sloane - what a lovely fellow. Great video.

[@JC-zw9vs]

More Neil please.

[@doctorphrog]

8:45-9:20 gave me chills.

[@deatho0ne587]

I have an issue with the inventory sequence, mostly in my head. -When he asked how many 0's in the second step there were 2, not 1 --1 count and 1 for number Yes, this would be a different pattern. Will use (count).(number), think the below is correct based on the same questioning and reason to go to the next line from the video. I am not sure which sequence would be correct though (also might be slightly off with both chain of thoughts) 0.0 2.0 0.1 1.2 0.3 5.0 2.1 3.2 2.3 0.4 1.5 0.6 etc... OR 0.0 2.0 0.1 5.0 2.1 3.2 2.3 0.4 etc...

[@Phriedah]

I can't be the only one who thought that the music felt really ominous in a cool way. Like, if I wanted background music for a haunted house, just play the first 10,000 terms in the series on loop over a speaker.

[@christopherhinzman7424]

Please do a video on the infinite sidewalk!! That’s fascinating. Thanks for sharing the link!

[@PhngluiMglwnafh]

I see a Neil Sloane video, I watch it, no questions asked

[@FlintStryker]

Always enjoy his videos. What truly amazes me though is there was a time when he consciously chose that wallpaper. 😂

[@WiseSquash]

5:36 amazing tune for a boss fight

[@krisrhodes5180]

"Using gahr-aage band yes" -- an epic moment of cultural history documented in this video

[@peligrosacurva-cz4ev]

It looks like SEE and WRITE 🎉

[@TimothyReeves]

Love it. He’s had that same stack of books to his right for a long time now….

[@sperenity5883]

God bless you, man.

[@davidbrooks2375]

The more we see of Neil's office, the cooler it gets!

[@LluviaSelenita]

I love these pieces of math art. I was hoping this would go towards music. It's awesome.

[@AbelShields]

So do you keep track of numbers bigger than 1 digit? So if there are 10 8s, does that get counted as 1 10 or 1 1 + 1 0?

[@andrewharrison8436]

This is a key comment, absoulutely right he isn't counting digits so far he is counting number of that size number, so if he was working in base 2, he would count 0, 1 , 10, 11, 100, 101 etc and get the same graph.

[@mattp1337]

I'm curious how the aspect ratio of each chunk (number of values n vs. the highest value) changes. Let's say we plot that (or some running average-like value) as a derivative sequence. It looks like it's rather stable, perhaps converging or oscillating around some value, but it's likely to be more interesting than that.

[@somebody2988]

I adore all of his video. He really makes math interesting, captivating and fun! I already dread for the day he shall pass.

[@patcheskipp]

It kind of sounds like the roar of a crowd that is in a panic. It gets excited and then the voices come to a murmur and then gets excited again. Or possibly a paniced or anxious mind

[@VladSuperKat]

With the 4:45 plot maybe you could improve the whale tail algorithm. With the 4:53 plot you can make organic shapes with it. Iguana backs, feet, Also with the 8:00 plot you can make landforms, also looks like torque curves when torque meets the horsepower so you could make car tunes with it :D It is a hidden gem of chaos theory :D You can use it anywhere almost like a logistic map.

[@peterdavidallison]

I for one would listen to an album length recording of the sequence on a grand piano.

[@mrmorganmusic]

Love this interview. One small note (ha): I wish his musical example had been Bach’s Goldberg Variations, which are themselves loaded with very purposeful mathematical design elements. Still, I appreciate a musical reference very much!

[@alexthebold]

Oh, this guy is great!

[@Algoritmarte]

Awesome sequence and wonderful explanation!

[@curtiswfranks]

Pick your favorite Mozart piece. What is the sequence which plays this musical piece perfectly on the piano via the OEIS mapping such that each term is positive but as small as possible?

[@user255]

I really like his videos! More!