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  • stern sequence.

  • A famous gutting and mathematician from the 19th century.

  • Here is his sequence.

  • It could be described in several ways.

  • It's very important.

  • It's one of the bigger entries in the only I s Let me give you the simplest definition.

  • The 1st 2 terms are one and one for the next two terms.

  • I copy the one in the one and put there some in the middle, so the sequence now begins.

  • 11121 And how does it continue?

  • I copy the previous line, and in between I put the someone Plus two is +31 plus two is three, so we have now shown you a triangle.

  • It's very like Pastor.

  • It's actually like theory.

  • Siri's f a r E y.

  • It's It's a very common number, theoretically.

  • So again, to get the next seven terms.

  • I copied the previous line, and in between them, I write some of the two terms about that, just as you said, just like Pascal's triangle.

  • Now this has a very nice property.

  • If I admit the last one, it has the property that if you look a ratios of successive terms, fractions one over 11 over to two over 11 over 3 1/3 insured three over to its one and 1/2 to over three.

  • We keep doing that.

  • We get all the fractions in their lowest terms.

  • It's a way of a numerator ng the fractions.

  • After a while, we get seven seventeenth's get 100 and one over 103 and so on.

  • We get 103 over seven.

  • We get all of them little ones and big ones, and they're always reduced.

  • So that's the secrets.

  • 112 1323 Just read it like you do with Pascal's trying off.

  • If you plot it, you get something that looks pretty nice.

  • Like I'm on a sketching the facades of some cathedrals.

  • This is half status, que sequence.

  • And the interesting thing is that we don't know if this exists or not.

  • It's an open question, Justus.

  • We don't know if Schrodinger's cat is alive or dead, Fibonacci says.

  • The day of N is equal to a of hen minus one plus Ava and minus to some of the two previous times off status sequences.

  • Event is a N minus a van minus one plus a of n minus A of N minus two.

  • Can you show me an example how that works?

  • Well, I guess All right, I'll try.

  • It begins with two ones.

  • Just like the standard naive for Ballouchy version.

  • We don't put a zero in here.

  • You go back one step, and you go back two steps, right?

  • That's how you get Fibonacci.

  • You look at where you are.

  • You go back one step and went back two steps and Adam up this one off.

  • Status que sequence you look at the previous term.

  • But that's how many terms you have to go back to get the thing you're adding.

  • So what's three?

  • So going back one step gives us a one going back.

  • One step because of that one gives us one.

  • One plus one is two.

  • So four.

  • What's therefore we look att.

  • This too.

  • So we go back two steps, two steps.

  • We get a one, we go two steps back and we get a one and that says, Go back one step and we get it too.

  • One plus two 03 So that one So so that one told us.

  • Go back one step.

  • But one step from our starting point again.

  • Yeah, And whatever you see on do your finger, you add it to the other one.

  • Let me see if I can get the next one.

  • Let me see if I can get him.

  • So for the fifth term, Yeah, that's a three.

  • So go back.

  • Three steps.

  • Yes, to three.

  • So little one.

  • Okay, so then we go back to numbers, which says, Go back to numbers to there.

  • And what do we see?

  • We see it too.

  • So we add that to the other.

  • Number six, get a three.

  • Let me see if I can get the next one.

  • Right.

  • That so the sixth.

  • Number 6 to 33 Bank 123 So give us a two, too.

  • And now 33 steps.

  • 1232 Again before and once more to make sure I've got a four steps.

  • 12322 and three.

  • Steps.

  • 123 for three.

  • Such a five.

  • Very good.

  • Okay, that's the sequence.

  • Well, now, if we're lucky, we can always do that.

  • But suppose there was a number here which was too big, and that that would say, Oh, you've got to go all the way back here where it's not defined.

  • If that happens, you're dead.

  • It hasn't happened yet, but we can't prove that it never happens.

  • Pretty obviously is never gonna happen.

  • If you look at the graph, this is 10,000 terms.

  • So most the terms are half.

  • Then we'd run into trouble if we had a term that was a big A Zen, because this line has slope 1/2.

  • Well, even though there is some little dots trying to escape some little particles trying to get away, they have to get it all the way up to here or something before they would kill it.

  • And it hasn't happened.

  • And it's not gonna happen.

  • We've checked Roman Piss computer 10 to the 10th terms.

  • Never, never a problem, but not proven now proven.

  • Here's a picture of 100 million terms that I got from Douglas Hofstadter, who invented the sequence and you can see beautiful ribbon like structure, but no proof it could die at any moment.

  • Beautiful.

  • You got another one.

  • It's a variant.

  • Slightly different rule, but uneven Maur striking picture.

  • Look at this ribbon like thing.

  • This one we know exists.

  • It's just a beautiful picture.

  • A beautiful graph.

  • What's the sequence?

  • That's a graph off.

  • It's called a chaotic cousin.

  • It's like the other one, but different.

  • And I'm not going to share the equation because it's an equation and you hate equations.

  • But it's similar.

  • Okay, okay.

  • I mean, it looks like that.

  • That's the definition.

  • This is a sequence that was invented by a colony.

  • Secrets.

  • It's a very simple rule, but little subtle.

  • You've got to put down a non negative number and always smallest possible.

  • So zero no one is gonna object to that.

  • No.

  • 12 If you look at them in by Nuri.

  • One looks like one and two looks like 10 and you notice there's no place where they both have a one one in binary and two and binary have diss joints.

  • Sets of ones.

  • They don't overlap one.

  • So there's no constraint.

  • As long as they don't overlap, you can have the same number.

  • Okay, now 3311 and binary.

  • And that overlaps both of these two.

  • So I can't put a zero here because we'd have to numbers that who's binary representations met that have the same number.

  • All right, so it's a what Now 44 is pretty good because it's not over.

  • Can overlap with anything, so I can put a 05101 Now I have to be careful.

  • I can't put a zero here because it would overlap with that and I can't put a one because three and five overlap below a story a bit.

  • So I can't put a one, but I can put it to six.

  • Is 110?

  • What can I put here?

  • I can't put a zero because of the two.

  • I can't put one.

  • I can't put a two because five and six overlap, so I have to put a three and that's it.

  • That's the sequence.

  • And what does it look like?

  • It looks like the Alps on a snowy day.

  • That's that's that's That's a stranger on that one.

  • Yeah, yeah, thank you for watching.

  • Also, a very special thanks to our patri on supporters.

  • Just some of them are the names you're seeing on the screen at the moment, and you can join them by going to patri on dot com slash number five.

  • This'll was the third and final in a trilogy of Amazing grafts, at least for now, until new shows us more.

  • If you'd like to watch the mole, there are links on the screen and in the video description, and we'll be back again soon with more number five videos.

  • You're probably wondering what film it actually looks like.

  • I wasthe yes, right from the start.

  • Here it is, it's pouring.

  • This is the first 200 terms and they go along and you know they get pretty big when we do these plots.

  • In the Hoyas, the first plot is 200 terms.

stern sequence.

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A2 初級

アメージンググラフIII - Numberphile (Amazing Graphs III - Numberphile)

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    林宜悉 に公開 2021 年 01 月 14 日
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