All right, With the first term is zero.

When you see a new number, the next term is zero.

So we saw a zero, but it was the 1st 1 So the next term is era.

We have seen the zero before.

We saw it one minute ago.

And when you've seen the number before, the next term is how far back?

How long ago was it since you saw it?

So we saw it One step back said the next term is one.

So now what's the next time?

Have we seen a one before?

No.

So the next term zero.

Have we seen a zero before?

Yes.

How long ago?

Two steps ago.

So the next term is too.

We've not seen a two before.

Right?

So we get a zero and we have seen a zero before.

It was two steps ago and we have seen a two before.

It was two steps ago.

So we get another two and we have seen a two before.

It was one step ago.

Now, for the first time, something really interesting is happening.

We have seen a one before, but it was 123456 steps back.

So we get six on.

We haven't seen a six before, obviously.

So the next term is zero.

Well, we've seen a zero before and it was 12345 steps back.

So we get a five for the first time.

So the next to him is a zero again.

And then we get to and so on.

That is Vanik sequence.

The obvious questions are what happens?

Do we see every number?

How fast does it grow?

What can we say about it?

We keep asking the same questions over and over again.

And this room always reminds me of a poem by Gene Beaumont, which is called Afraid so.

And it sort of goes like this.

Is it starting to rain?

I'm afraid so.

Is this going to hurt?

Are we out of coffee?

Is the car totaled?

Will this leave a scar?

And so on?

The title is afraid So I was thinking There's so many questions.

We ask about the sequences.

So the poem here would be called Don't know What's the next time?

I don't know.

How fast does it grow?

Don't know.

There's every number appear when someone just comes up with a court sequence like this and there's all these.

Don't knows who's drop, is it to find out What is it?

You know what happens.

Well, this is one of the great things about being an editor of the Oh, yes, this sequence came in.

It was a submission on one of the pleasures.

The editors don't get paid, but they do get to see these sequences when they come in for the first time.

And so you look at him, you say, Boy, that's a really great sequence.

I'm gonna analyze that before somebody else does.

You see so many sequences in it.

Why is this one make you smile like that?

Yeah, I think it's lovely.

Well, partly because it's it's really quite unpredictable.

And so, of course, one of the first things you D'oh!

If you look at the graph, let me see if I can give you a sketch.

So there's zero and zero and a one and zero and two.

But basically what happens?

It goes and it drops down to zero and it grows and so on.

And if you look at a lot of terms intends to look a bit like this if you close your eyes and squint, if sort of fills up the whole try, and it seems that it's growing about linearly.

If you look at this line, goes through the top, it seems to have Slope one.

In other words, the instruments about N.

But we can't prove that has fun of the many things we can't prove about this.

There is one thing we can prove.

It keeps growing.

It could be that after a while we don't see any new numbers.

They're no more zeros.

If the maximum number that ever appeared was 100 then after we seen 100 will be any more zeros because there won't be any more new numbers.

And the argument is very, very pretty.

You say?

Suppose the some number.

Let's cool it, M.

Maybe it's 100.

Or maybe it's a 1,000,000 which is the maximum number that appears ever in this sequence.

Now, what that would mean is, since the sequence is defined by how far back you look for the last term, that if the miss number is, say, 100 we would never have to look back more than 100 terms in order to know the complete history.

From that point on, everything would be determined.

But then you say, If we only have to look back em toe or n plus one terms and each of those terms is in the range of one to em, then there would only be M to the M possible terms.

Supposing that there is a biggest number at we look a chunk of length M that determines everything from that point on.

But then you say there are only em to the M different chunks, so one of them will have to repeat, so the sequence will be periodic.

So from a certain point on, the sequence will repeat.

But that contact I want to see the proof.

I think it's quite nice.

So here's the sequence, and this part is the period.

And then it repeats.

From then on, it just keeps repeating over and over again and let me prove that can't happen.

Let's look at this chunk.

Say it begins with ex dot, dot, dot and it ends with a Z in the next period.

It's the same thing.

Just repeats ecstasy and so on.

There might be other copies of Z in the period.

So let's look at the first time we see a Z.

So this is the first copy of Z in the period.

And let's look at the term that comes after it on.

Let's cool.

It's a now.

What that means is, by the rule, the definition of the sequence.

Is this a means?

Last time he saw the Z was a steps back.

That's what this a means.

OK, now let's move to the previous period.

So we have the Z and we have the eh It's exactly the same as this, but this a means that this Z actually appeared a steps back.

So it was here.

So that means the period didn't begin there the first period it began one step earlier, which is a contradiction, so that can't happen.

So the sequence keeps growing, and that's about the only thing we know about it.

There's no maximum number that way.

We know for sure there's no maximum number.

There are more and more zeros beyond that.

We know almost nothing, and there are a number of other questions I could I should mention.

I mean, for instance, does every number appear?

I don't know.

Okay, folks, here's one for you.

How many ways can you arrange plus and minus signs between these numbers and get a result that's divisible by three.

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