Dungeon Numbers - Numberphile

[@PTFVBVB]

There's something about Neil's voice that has a "teacher that really cares about your learning" quality to it

[@sillysausage4549]

Strange. I find his corrupted English accent incredibly annoying. Sure he's a nice bloke, but the American pronounciations really grate on me.

[@Liveitlarge247]

He has a wicked T-shirt on too

[@leif1075]

For me a soothing sensual therapeutic quality!!!

[@Nihilistless]

I completely agree. He seems earnest.

[@Bronco541]

@@sillysausage4549 I like how his accent is somewhere between English and American, its unique (somewhat, I'm sure there's plenty of people who move around a lot with similar accents). But more than that; he's enthusiastic and passionate about what he teaches.

[@homegronhomestead8640]

You should switch from calling it 'dungeons' to 'BASEments'

[@jocabulous]

badum tss

[@david_ga8490]

R/whoshhhh

[@skrrrrrrrrt]

David Gallego Álvarez huh

[@zegichiban]

I thought I like it being called dungeons, until I saw this comment.....

[@sallylauper8222]

aLL YOUR BASEMENT ARE BELONG TO US.

[@Henrix1998]

How about ... 12 11 10 11 12 ...

[@patrickhanlon932]

It all depends on how you parenthesize it.

[@jeremydavis3631]

EDIT: This was my first impression. I've made another comment after thinking about it a bit more. Given that the size of a tower is unbounded and the size of a dungeon is asymptotically bounded (so that we only need two nested logarithms to get it to a friendly size), the combination should diverge to infinity, just slightly more slowly than the tower alone would. Calculating the terms of the sequence would be quite a bit harder, though.

[@ShevkoMore]

............... 12 12.. 11 11.. 10 10 10.. 11 11 11.. 12 12 12.. ...................

[@jeremydavis3631]

Oh, wait, I think my first impression was wrong. As Patrick Hanlon said, it depends on how you parenthesize it. If the last operation we do (assuming it makes any sense to talk about a "last" operation in an infinite sequence) is part of the tower, the result should be unbounded, since we're raising a large number to an unbounded power. If the last thing we do is part of the dungeon, it'll drag us back to the asymptote, and so it *might* be bounded. Proving that it actually *is* bounded for any given parenthesization strategy doesn't seem easy, though.

[@henryirvine7964]

no

[@otakuribo]

There's a valuable treasure awaiting brave adventurers at the bottom of this dungeon, and his name is Neil Sloane

[@Triantalex]

false.

[@Adam-ds1ik]

Props to the editors and animators. This was pretty dense but their help made it understandable

[@unnamed7225]

2:22 Took me a while to figure out that 11 was actually an equal sign rotated 270 degrees. because who says 90 degrees these days

[@ChadTanker]

every body does say 90 degrees because its shorter

[@unnamed7225]

@@ChadTanker Then what should everybody say for negative 90 degrees?

[@cubixthree3495]

@@unnamed7225 negative 90 degrees

[@unnamed7225]

@@cubixthree3495 ;-;

[@wolfiy]

@@cubixthree3495 3pi/2

[@w4yland3r27]

If you listen to this without watching, it's like a madman just rattling off numbers.

[@kasperrosenlund4187]

That would describe a lot of Numberphile videos :D

[@Vgamer311]

It’s like that if you’re watching too.

[@andybaldman]

@@Vgamer311 lol!

[@hayuseen6683]

It stopped making sense after 40 seconds in, after that it was Numbers Station ramblings.

[@not2tired]

This also works if you watch without listening

[@ncot_tech]

New mathematical terms here - “pretty big”, “gigantic” and “really tiny”.

[@Melomathics]

We sometimes use this kind of terminology. Others include: almost everywhere, almost surely, almost never, always never, etc...

[@duskyrc1373]

And all three terms can apply to the same number, depending on context

[@andrewboyd9948]

@@duskyrc1373 bruh

[@BoundlessxArts]

@@Melomathics "70% of the time it works every time"

[@JobvanderZwan]

Nothing will ever top "the tooth number" though

[@captdeadfool5685]

I got no idea wtf you're talking about but i like how you write stuffs on that brown papers

[@binodbinod6814]

Binod

[@EHMM]

E

[@sbmathsyt5306]

Never heard of this but that is what is so great about this channel, always bringing fascinating new concepts to the viewers attention. This has certainly inspired me to look more into different bases.

[@abogmus8904]

Neil sounds like a Half Life scientist

[@zanvure330]

Lol well said

[@jschoete3430]

How is that so precise haha

[@ericschuster2680]

the test chamberrrrrrrrr

[@ionymous6733]

i hear Professor Farnsworth from Futurama

[@carltonleboss]

He was the G-Man the entire time

[@JSLing-vv5go]

Sloane is great. I love integer sequences.

[@chrisdoyle1389]

He's a Psycodelic Maths Professor,He has a Hendrix T-shirt on.

[@christopherellis2663]

@@chrisdoyle1389 psychedelic

[@BrianShelfPartTwo]

Every time I see Mr Sloane's videos I can't take my eyes off his folders. Please can I ask, what are "Fat Struts" ? Thanks for the content y'all.

[@eac-ox2ly]

True

[@lawrencecalablaster568]

Apparently they’re a mathematical structure in a lattice about which he has written a paper!

[@Triantalex]

??

[@PC_Simo]

5:10 It’s just 10 (or whatever your original base / starting number is) + T_n (the nth triangular number).

[@LeoStaley]

If number explanations at the online encyclopedia of integer sequences (oeis) were like this, I would spend more time exploring it.

[@noidea2568]

At first I was like "wait, this is a really simple pattern, 10, 11, 13, 16, 20... that just means that I have to add 1 the first time, 2 the second time, 3 the third time and so on and so on". But then I saw the numbers at 6:13. Oh boy was I wrong. This pattern is not as simple as I thought.

[@martind2520]

The third sequence does actually follow that pattern, so you weren't completely wrong.

[@KnakuanaRka]

Yeah, I think it starts like that, but I believe it stops working once you get beyond 20.

[@-johnny-deep-]

Yeah. Surprised that wasn't pointed out in the video.

[@dustysparks]

So the "magic jump" in these sequences happens when the second units number increases from 1 to 2 (ie "10 sub x" to "20 sub x")

[@rosiefay7283]

Which shows just how fundamentally bogus this whole setup is. It confuses numbers with decimal representations.

[@menachemsalomon]

@@rosiefay7283 No, I don't think that's so. Firstly, when we're discussing different bases, only the first step is decimal. But Dustin is saying that the jump happens when the second place (not the units, the n^0 place, but one to the left, the n^1 place) goes to 2.

[@Martykun36]

​@@rosiefay7283 not really? it just means you have to settle on some "global base" first, and in this case it was 10. You can do the same process for any other base.

[@JovianCloudfarmer]

This does still end up pretty base 10-centric, even though it plays with many different bases. I looked a little into how it ends up when you keep it all in binary and only convert to base 10 at the very end, and it was pretty interesting, since for example, the 4th step is no longer 10_11_12_13, it's 10_11_100_101. The introduction of a third digit in the base so quickly means that you start to square numbers in the base conversion process sooner, so the numbers start to grow bigger sooner. However, since it's powers of 2 and not powers of 10, I suspect that the size of the growth rate changes will be smaller, so it's very possible that base 10 will catch up in terms of number size after a number of steps. An example (using bottom-up parentheses): Base 10, 7th step: 10_11_12_13_14_15_16 = 31 Base 2, 7th step: 10_11_100_101_110_111_1000 = A 68-digit binary number, 193825204350418564226 in base 10

[@Npvsp]

His voice and tone are so relaxing and mesmerising!!

[@dejremi8190]

If you love Neil sloane's numberphile videos, clap your hands (clap clap)

[@LeoStaley]

Clap clap! 👏

[@emdivine]

clap clap

[@Sci0927]

clap

[@yashrawat9409]

Clap

[@verypotato6699]

*Clapping intensifies*

[@FandangoJepZ]

I love how I was so fooled by the first dungeon sequence. I compete in a lot of math competitions so I got very full of myself, and it was obvious the increment was always increasing by 1 and then it went like NOPE

[@lawrencecalablaster568]

It’s so strange how it fits exactly up to 65 & then exponentially increases.

[@19Szabolcs91]

@@lawrencecalablaster568 Sure is, but it has to do with how the difference in the sequence gets to be a 2-digit number, breaking the pattern. Similarly, the reason all 4 sequences started with 10, 11, 13, 16, 20 comes down to "'1" being the first digit.

[@selseyonetwenty4631]

Well exactly, that's what I would like to see a focus on. What are the break points and why are they where they are? Did he ever get to explaining that? I bailed out I'm afraid, first time I have done that in a Numberphile video. Seeing a million conversions of number base one after the other was ... let's say tedious.

[@nilsragnar1347]

Neil Sloane might be my favorite guest on Numberphile, glad to have him back!

[@GhilesNc]

7:01 : You forgot the paper change music !

[@david_ga8490]

Yep

[@bobbyyie1310]

@@david_ga8490 you don't want to attract headless creatures and such whilst in a dungeon.

[@Decessus117]

At first I was surprised by the growth of these sequences. However, after some thought, I think there's an intuition to be had here. When interpreting a number in a base (e.g., interpreting 153 in base 10), you *are* performing an exponentiation in some sense, because you're interpreting it as 1x10^2 + 5x10^1 + 3x10^0. But the trick here is that, despite interpreting the numbers in all these different bases, *we are restricting ourselves to the 10 regular digits!* So unlike in, say, hexadecimal, where the number after 99 is 9A, here the number after 99 is still 100. As a result, the instant that one of these sequences increments its second term, or reaches a 3rd term, it starts to grow by a factor of the base (and the base has been increasing for some time). This helps it very quickly reach a fourth term, and thus grow by the cube of the base, etc. After that it's clear to see why it explodes. If we allowed as many digits as bases (e.g., 8, 9, A, B, ...), the terms would just grow by one each time and the sequence would stick to the triangular numbers.

[@vsm1456]

oh, that's cool

[@adizmal]

When you cross that threshold of having no idea what's going on, but there's still more than 10 minutes left in the video...

[@Maharani1991]

Hahaha :D

[@Terri_MacKay]

I got to the point where I was beginning to catch on...then he started talking about logs, and I was completely lost again. I am terrible at math, but find it fascinating. I understand a lot of the videos on this channel, but some just go right over my head.

[@penfold-55]

And then he starts referring to dollars!

[@Terri_MacKay]

@@penfold-55 Yeah...what was that about?? Is "dollars" a math term I don't know about?? 🤔😂

[@-johnny-deep-]

@@penfold-55 - Yeah, that was odd. I guess he thought it would help people understand. I was understanding great until he temporarily threw me by saying "dollars" :-)

[@Hyo9000]

I love Neil Sloane, he’s becoming one of my favorite Numberphile hosts

[@efa666]

Why does this guys office look like the inside of a circus tent?

[@lukefreeman828]

You mean "why do circus tents style themselves on this guys office?"

[@HitHard1008]

@@lukefreeman828 stop there.

[@earthwormscrawl]

Is it an office raised to the power of a circus tent, or a circus tent in base office?

[@Eric4372]

It’s the Whataburger wallpaper 😂

[@omikronweapon]

actually, this is the fírst time I realised he's just in a room with stripey wallpaper. My mind always interpreted it as him being in a tent, at some mathematical excavation xD I never questioned it...

[@Playmaker6174]

Yesss, more Neil Sloane and numbers :)

[@cassa995]

This video just shows how to get the sequence 10 11 13 16 20 from various different methods

[@Meuszik]

AND how using those methods produce radically different divergences _after_ 20.

[@OKRASSnaky]

Ok, neat to follow until... Wait, what? 1.1? a non-integer base?! :o

[@Jordan-zk2wd]

(you can even have imaginary and complex bases actually ^ ^)

[@MrAlRats]

There are numeral systems that use complex numbers as their base. For example, the Quater-imaginary numeral system which uses the imaginary number 2i as its base. It is able to almost uniquely represent every complex number using only the digits 0, 1, 2, and 3. No minus sign is used for negative numbers in this numeral system, as they have a different representation from their positive counterparts.

[@Lightning_Lance]

This is a delicious irony because the word dungeon comes from donjon, which was the main tower in a castle.

[@vmp916]

Every year, my local university in NJ has a festival that features lots of school clubs, departments, and occasionally artists, researchers, vendors etc. I first met Neil at one of these special days. He had a table set up with sequences as puzzles where you had to figure out the next number and what the sequence was. If you were interested, he would talk to you about more sequences and the OEIS. I met him again another year. To my knowledge he is a regular attendee. Obviously they didn’t have any festival day this year. It’s a treat getting to see him talk about interesting sequences in video form regardless.

[@CoolAsFreya]

I can't help but grin at the absurdity of the sequences that mathematicians come up with

[@teslapenguin1]

I’ve heard about sub used for counting variables (a1, a2, a3, etc), where a1 is term 1, a2 for term 2, etc. but I haven’t heard sub used this way.

[@awayname5008]

You can´t just leave on a cliffhanger like that.

[@linggamusroji227]

Your shirt looks great, we both love Jimi Hendrix

[@azhakabad4229]

All amazing stuff is here!

[@unnamed7225]

I realized that when you did the example for top to bottom and showed the sequence, I noticed something... I am just commenting right after seeing it so I don't know if you mentioned it in the video but... The sequence is 10, 11, 13, 16, 20, 25, 31, 38... I noticed the sequence is 10, then 10+1, then 10+2, 10+3...

[@thomasbui6175]

I noticed at the first way of bracketing, it is just +1,+2,+3,+4,etc. But top down it changes after the +4. That's a cool pattern.

[@ericschuster2680]

Is this the guy who knows the plot and character names of Avatar? What a legend!

[@kseliascryser5259]

7:25 casual explanation that 11 base 10 is indeed 11 base 10 :D

[@davidgillies620]

The first sequence is A121263 in the OEIS. In Mathematica: define the rebase function, rebase[v_] := Join[Drop[v, -2], {FromDigits[IntegerDigits[v[[-2]]], Last[v]]}] Then define the dungeon number function to apply this recursively to a list of numbers: dun[n_] := First[Nest[rebase, Range[10, 9 + n], n - 1]]. Now make a table: dun[#] & /@ Range[20] which gives {10, 11, 13, 16, 20, 25, 31, 38, 46, 55, 65, 87, 135, 239, 463, 943, 1967, 4143, 8751, 18479}.

[@empty5013]

love neil's videos every time, this man is the integer wizard

[@Stemma3]

I barely understand the theory but watching Sloane having fun with secuences is awesome.

[@SolomonUcko]

5:26 This relies on converting to decimal before reinterpreting it in the target base, the sequence would presumably be different if calculated using another base.

[@Joe-wj7ku]

I've wondered about the order of indeces since I was in high school. I'm so grateful I've found a video about it!

[@frogandspanner]

It's good to see I am not alone in my filing system, especially the heap of books (One of my heaps at home became unstable, collapsed, and broke a table!) I extend the heap system thus: 1) Place anything incoming on one of the heaps on my desk 2) When needed, search for the item in the heap and, when finished with it return it to the top of the heap. 3) When the heaps become too tall to see over a) Take off the top half b) scoop off the bottom half into the bin c) Return the top half. In that way the communication from the Vice Chancellor progresses at a steady pace to the bottom of the heap and to the destination it ultimately deserves.

[@LunchboxGaming]

5:00 Is it just a coincidence that if you add the right most digits of the descending numbers to the top number you get the end number. (not a math whiz)

[@n0t10c]

I was just coming here to post this

[@SgtSupaman]

No, it isn't a coincidence, because the left digit is a one, which means it is just equal to whatever the base is while the right digit is equal to itself, so you are adding the base + right digit. Calculating bases (which essentially means converting from whatever base into base 10) looks like this x^y*a + x^(y+1)*b + x^(y+2)*c + ... (where x is the base, y=0 because you are starting from the first position left of the decimal, and [a,b,c,...]=whatever value is in that position). It is the same thing that you learn as a child when you say a number like 13790 has a 1 in the 'ten thousands' place, a 3 in the 'thousands' place, a 7 in the 'hundreds' place, a 9 in the 'tens' place, and a 0 in the 'ones' place. That means that the number is equal to 10^0*0+10^1*9+10^2*7+10^3*3+10^4*1 (or 0+90+700+3000+10000).

[@LunchboxGaming]

@@SgtSupaman Word...

[@ChavvyChannel]

With every episode is even more and more effort for the animations

[@carpediemcotidiem]

Love this guy's passion for his subject

[@Zheunchain]

There seems to be a mistake on the brown paper at 9:53 Neil skipped 19 in base 14 and went straight to 19 base 13. the result should be 28 not 27.

[@GenericInternetter]

he also made a mistake in the introduction, where he did 12*5 instead of 12^5

[@vladislav_sidorenko]

@@GenericInternetter That is not a mistake. (a^b)^c = a^(b*c).

[@esotericVideos]

It's interesting watching numberphile and getting a sense of the different mathematicians personalities. Some of them really like working towards some theory, some like real world implications, some like "giving it a go", and some like Klein bottles. But Neil Sloane more than anything seems to just like to play with numbers. There doesn't seem to need to be any greater meaning than saying "what if we play with weird rule X with these numbers". It makes sense why such a personality would create the OEIS.

[@mostlyokay]

I can't help but get a little dumbfounded by videos where he appears precisely because of that. In my mind here is no point in just finding number sequences without any connection to anything else in maths. But of course, time and time again results that were thought to be purely abstract and disjointed from other fields of maths have proven to be just the opposite.

[@rangerocket9453]

3:57 - 4:00 I died of laughter Neil: plus 2 **awkward pause** uh - au - um dollars Me: *[Breaks into Laughter]*

[@dieselguitar1440]

Wow, that's amazing! I thought that it was just a boring quadratic at first, and would've passed it off as such if it weren't for this video showing the cases past only a few iterations. What's going on here (I think), is that the bottom up approach starts getting "faster" with more digits, and the top down approach starts getting faster once to 10+X turns into 20+X.

[@aaroncarsonart]

10:02 I am delighted that the first five numbers of all 4 sequences are 10, 11, 13, 16, 20. I'm also appreciating that for two of the sequences the differences of sequential elements continue to be the natural numbers for a while longer.

[@manuelsaavedraabarca9318]

Sloane's videos are my favorites

[@TyTheRegularMan]

It's fascinating that all these sequences start with the same exact five numbers before diverging.

[@albinoasesino]

7:00 "...natural way to make a dungeon. If you give me a bit of paper I'll show you." Taken out of context, it would sound like Neil is trying to get fundings for this esoteric long staircase just going up, another even longer coming down, and one more leading nowhere just for show.

[@ChukapiMagnetar]

9:54 Brady and Neil got different answers... Which really emphasizes how math can be more slippery than metal on ice

[@thedystopyansociety]

27 seems to be correct in this case. Drove me a bit mad trying to figure out how they arrived at 28 in the graphic.

[@PeridotFacet-FLCut-XG-og1xx]

If you go from top to bottom, we're writing it in base ten (decimal), wouldn't this affect something? If you go bottom to top, it doesn't matter because we only care for the value.

[@InigoSJ]

He's back! Thaaaanks so much, more ASMR for me to sleep.

[@Klaevin]

what program do you use that gives all the digits for huge numbers to put in the video?

[@BryanWLepore]

A visit with Neil Sloane is a great way to lift our mathematical spirits out of the dungeons, for sure.

[@sm64guy28]

There are two kinds of numberphile videos, either « the next number in the sequence is really big » or the « we still don’t know if the next number in the sequence exists, we’ve checked up to numbers that are xxx digits long »

[@businessguide6219]

Officially, you're one of my favorite KZbinrs out here!

[@Naokarma]

To fix the ambiguity of the towering numbers, this is why we need the triangle of power, which replaces exponents, logs, and roots with a single notation, and shows no ambiguity for things like this, as well as more clearly showing the relationship between the 3 notations. For those who don't know what this notation is, 3Blue1Brown did a fantastic video on it, and I highly recommend anyone watch it.

[@rosiefay7283]

3:05 This is reasonable. The "rebasing" operation treats its first (top, left) operand as a digit-string, and evaluates it in the base given by its second (bottom, right) operand, and gives you a number. So anything with a subscript is a digit-string, not a number. So in a stack of dungeons every level is a digit-string except the bottom one, so you have to start at the bottom and work up.

[@SquirrelASMR]

Can u get more of this guy and OEIS and Amazing graphs?

[@tal4726]

When you're watching a bunch of videos on Dungeons and Dragons and your recommendations get a little weird. Hi, I wasn't expecting this but this channel seems fun

[@Xonatron]

5:56 - great visual animation here!

[@gamespotlive3673]

This is really cool. Like a entirely new way of thinking about numbers.

[@gamespotlive3673]

I don't know why I wrote this.

[@lawrencedoliveiro9104]

10:41 So towers are clearly made out of timber, since you can take them apart log by log. ∗Ahem∗

[@a.a7907]

Thanks for your video. If you can share a complete course about what is electricity and how to manipulate it. What are some useful devices that every system must have. How to make projects out of these devices. This would be great thing to have.

[@originalveghead]

I enjoyed this video way more than I probably should have.

[@Aleph0]

i skipped to 10:57 and my soul nearly flew out of my body

[@michakuczynski2987]

Neil Sloane is by far my favourite guest on Numberphile :)

[@The_Commandblock]

Arent the first just 10 + a triangular number

[@sharcc2511]

This video taught me how to count in bases higher than base 10, despite that not being it's main goal.

[@MyYTwatcher]

Interesting is that his brown papers are awesome, but my brown papers are shitty.

[@B1GB1RDB4G3L]

Omg I love videos with Neil

[@Vgamer311]

I don’t think this was addressed (or I just missed it) but in the sequence 10 9 8 7... With parentheses starting at the top, it’s not even possible to have an infinite sequence because before long the number being operated on will contain digits not defined in the base being converted to. It’s like saying 5 base 2.

[@Axacqk]

Is the "slow" growing parenthesizing starts as a quadratic function because there are two digits, so the second power of the new base is the largest that ever gets accumulated into the next number in the sequence. But the moment the sequence reaches three digits, suddenly the third power of each consecutive new base comes into play. That causes four digit numbers to be reached even faster, and then it explodes.

[@trummler4100]

I've found a list of Triples and the first 10 are the following: 20, 15, 12 156, 65, 60 255, 136, 120 600, 175, 168 609, 580, 420 1295, 444, 420 1640, 369, 360 2385, 1484, 1260 3660, 671, 660 4015, 3504, 2640 What might those triples have in common? :) It would be awesome, if you could create a video about those triples, +Numberphile :D As soon as someone found out what those triples are, I'll post a python code below to find those triples ^^

[@noonenothing422]

Why can't negative numbers be considered prime? I understand that to classify a negative as a prime, this would interfere with already established theorems and axioms; but what about giving negative primes their own classification under for n

[@WRSomsky]

One oddity w/ a "base computation" (a sub b) is that 'a' *isn't* really a numerical value, but a character string. If you do a top-down, you're constantly having these "represent in base ten" conversions.

[@jodfrut771]

Neil is always great

[@ThePaci93]

I love this channel

[@Endureth]

Quickest I've ever gotten lost on a Numberphile video!

[@secularmonk5176]

9:04 Given how much "11" figures in this exploration, his rotated "=" is triggering me ...

[@joedeshon]

Great video, as usual. But I missed the requisite elevator music during the paper change at 7:01.

[@gurrrn1102]

The first few minutes of this video were as if Fermat had found an elaborate way to generate the triangular numbers.

[@royalninja2823]

Huh, I honestly didn't expect the sequences to grow that quickly, I noticed that consecutive digits were just consecutive digits apart, i.e. 10, + 1 = 11, + 2 = 13, + 3 = 16, + 4 = 20, etc. Expected that trend to continue but didn't take into account how 3 digit numbers would be interpreted wildly differently.

[@glowstonelovepad9294]

Up to 20, they are 10 + the triangle numbers.

[@bdtv463]

Dont forget to place torches when you dig that deep

[@vijaykumarpawar1951]

Can someone please proof this q . Tan(3π/11) + Sin(π/11)= √11. Please try to do it ASAP

[@vijaykumarpawar1951]

It is hard.

[@KipIngram]

I'm a little unsure which parenthesis default like better on exponents, but it just feels clearly to me that in the dungeon numbers you should work from the bottom up. NvM (N sub M) CLEARLY is ONE NUMBER to me. In NvMvK that MvK means something, and in particular it means "M in base K," so using M in any other base just seems completely wrong to me.

[@williamcollins4049]

Best use of the brown paper yet.

[@ditrixgenesis781]

For anyone who hasn't gotten into base conversions yet, there's a confusing line in here. Single digits can change, if the base you're converting to is smaller than the digit. This was shown when he wrote 7 in binary

[@nutsnproud6932]

I wish Bill was my maths teacher. Thanks for the video.

[@matematixyt]

anyone noticed that the first and third sequences resulted in a list of triangular numbers + 10? 10 - 10 = 0 11 - 10 = 1 13 - 10 = 3 16 - 10 = 6 20 - 10 = 10 25 - 10 = 15