0:00:00.240,0:00:05.040
Sometimes, when you're trying to come up with a 
sequence or just looking into patterns and things,

0:00:05.040,0:00:10.880
you find a rule and it turns out there is no next 
option. The rule you wanted just doesn't work.

0:00:10.880,0:00:16.640
Here's a sequence I found in the OEIS - the 
Online Encyclopedia of Integer Sequences - that

0:00:16.640,0:00:23.680
grabbed my eye, because it's called "the 
infinite trunk of least squares beanstalk".

0:00:23.680,0:00:26.640
And there's really not much 
more explanation than that,

0:00:26.640,0:00:32.640
which is really... if that doesn't pique your 
curiosity, what will? It says it's the only

0:00:32.640,0:00:37.280
infinite sequence - we haven't learnt anything new 
yet, it said infinite at the beginning - such that

0:00:37.280,0:00:47.120
a(0) is zero and a(n-1) is a(n) minus the 
least number of squares that sum to a(n).

0:00:47.840,0:00:53.120
So, it said 'least squares beanstalk'. We've 
got the least number of squares again. And

0:00:53.120,0:00:59.360
then there's no other explanation about 
what that means, how you work it out. Oof!

0:00:59.360,0:01:05.120
I've just spent a few minutes doodling, working 
out what that means, and it's actually quite nice!

0:01:05.120,0:01:10.400
One of the problem with the OEIS is, because there 
isn't much space for explaining what you mean,

0:01:10.400,0:01:17.280
the titles tend to be a bit oblique, like this 
one, and then you'd want a bit more motivation,

0:01:17.280,0:01:21.600
explanation, links to things. I think the 
assumption is that you will have published

0:01:21.600,0:01:25.920
something somewhere else about it. 
But no such link on this entry.

0:01:25.920,0:01:40.160
So, the sequence is - and I'll copy it down - 
A276573. If you're watching this when the OEIS

0:01:40.160,0:01:44.720
has moved to seven digits, hello person from 
the future! I wonder how far away that is.

0:01:44.720,0:01:54.880
And it goes 0, 3, 6, 8, 11, 15, 16, 
18, 21, ... and we'll stop there.

0:01:54.880,0:01:59.760
OK, so it said it's infinite. My dot-dot-dot 
is correct there, it's going to go on forever.

0:01:59.760,0:02:03.680
Where do these numbers come from? What 
do they mean? Where's the beanstalk?

0:02:03.680,0:02:07.280
So it says it's the only infinite 
sequence such that this particular

0:02:07.280,0:02:11.440
rule is followed. So maybe there are 
other ways of trying to follow this

0:02:11.440,0:02:16.960
rule that make a not infinite 
sequence. So that's intriguing.

0:02:16.960,0:02:24.800
First thing to try out is - it goes 0, 3, so 
start going 0, 1 must stop. So let's try that!

0:02:24.800,0:02:28.960
So I'll write my zero. We'll start 
with... let's try and make it a beanstalk,

0:02:28.960,0:02:35.280
eh? So write a zero here. I'm going to 
try and go upwards, like beanstalks do.

0:02:35.280,0:02:40.720
Up to 1. So does that obey the rule? What's 
the rule again? I'll write it down. a(n-1)

0:02:42.000,0:02:49.200
is a(n) minus least number of squares, 
which is another sequence in the OEIS,

0:02:49.200,0:02:56.960
so I'll give it its number. 
A002828. Nice number! Of a(n).

0:02:58.720,0:03:06.560
Right, so I know a(n-1). That's my zero. 
And I need that to... I need an a(n) that

0:03:06.560,0:03:14.320
satisfies this rule, so a next term. So does 1 
satisfy it? So, if a(n) is 1, the least number

0:03:14.320,0:03:22.720
of square numbers that sum up to that is 1. 1 
is a square number, 1 adds up to 1. OK, cool.

0:03:22.720,0:03:26.640
So 1 - 1 = 0. Bingo.

0:03:27.440,0:03:38.720
Now next, I need to find another number for 
my a(2)... so that's a(0), that's a(1). I need

0:03:38.720,0:03:46.480
a(2) to be something that if I take away the 
number of squares that add up to it, I get 1.

0:03:46.480,0:03:50.080
I don't know off the top of my head 
how many squares you need to add up

0:03:50.080,0:03:56.960
to certain numbers. We could try working a 
few out, I suppose. So I'll go over here...

0:03:56.960,0:04:02.680
We've got n and A002828 of n.

0:04:02.680,0:04:03.600
Bloop!

0:04:03.600,0:04:08.880
Right, so zero. How many squares add up to 
zero? If I add up zero squares then I get zero.

0:04:08.880,0:04:17.600
One, one. Two, um... I need two squares. 
One plus one. Three, I need three squares.

0:04:17.600,0:04:23.360
Four, that's a square number. I only need 
one square number. Four's a square number.

0:04:23.360,0:04:30.320
Five is four plus one, so that's two. Six is 
four plus one plus one, that's three. Seven,

0:04:32.160,0:04:34.613
four plus one plus one plus one.

0:04:34.613,0:04:36.160
For eight, four plus four, two squares!

0:04:36.160,0:04:39.040
So, right, this carries on forever.

0:04:40.720,0:04:49.600
I need a number such that the difference 
between these two things is one.

0:04:49.600,0:04:53.360
Can you see one?

0:04:53.360,0:04:54.160
I can't.

0:04:54.960,0:04:58.560
Now, cleverly, I've gone off the edge 
of shot there, so I'm going to go over

0:04:58.560,0:05:02.560
to the left here. So this is the difference 
over here. I'll write down the differences.

0:05:02.560,0:05:08.560
0, 0, 0, 0, 3, 3, 3, 3, 6.

0:05:08.560,0:05:13.920
A person might be inclined to think at this 
point, does this thing always go up? Certainly

0:05:13.920,0:05:17.600
not going to come back down to one again because 
it's a theorem that every number can be written

0:05:17.600,0:05:23.840
as the sum of at most four square numbers. 
OEIS says that Lagrange came up with that.

0:05:23.840,0:05:28.720
Nothing's going to have a 1 in this column, 
which is what I want. So 1 doesn't work. That

0:05:28.720,0:05:35.440
is a finite beanstalk, which means 
I've got to rub it out. Excuse me.

0:05:35.440,0:05:38.640
So, we're getting there. We can 
rule out two by the same logic.

0:05:39.440,0:05:44.400
Can't see any twos in this column, 
it's not coming back down to two.

0:05:44.400,0:05:51.360
Shall we go three? Three's the last choice I 
could've had, because it's the last thing where

0:05:51.360,0:05:57.360
the difference is zero. So in order for this 
to work at all, it's going to have to be three.

0:05:57.360,0:06:02.080
Sometimes, when you're trying to come up with a 
sequence or just looking into patterns and things,

0:06:02.080,0:06:10.720
you find a rule and it turns out there is no next 
option. The rule you wanted just doesn't work.

0:06:10.720,0:06:14.000
Right.
So next term's three.

0:06:14.000,0:06:17.120
That's my a(1).
So again, next,

0:06:17.120,0:06:19.840
I've got to choose the next term. What's my a(2)?

0:06:20.560,0:06:29.360
So I want something that when you take away the 
number of squares counting function you get three.

0:06:29.360,0:06:33.760
So I want something with a three in this 
difference column. My choices are four, five

0:06:33.760,0:06:42.960
And, yet again, I'm going to use the 
fact that this differences sequence,

0:06:42.960,0:06:46.400
I don't think decreases. I'm 
pretty sure it doesn't decrease.

0:06:46.400,0:06:52.000
So four is out, five is out, six will work.
So just so you can -- ooh, hang on.

0:06:52.000,0:07:01.680
So a(1) is three. So a(2)... 
three equals six minus,

0:07:01.680,0:07:05.040
that thing over there, the 
thing in the six column. Three.

0:07:05.040,0:07:08.320
So three is six minus three. Works. Bingo.

0:07:08.320,0:07:13.520
I've got, I think, a rough idea of how this 
thing works. So if I keep extending this table

0:07:13.520,0:07:21.040
down and I use the rule of, I try and look in 
the column here for the last term I had, then

0:07:21.760,0:07:25.360
hopefully I'll always find something that works.

0:07:25.360,0:07:27.760
I'll quite often have a few choices. So some of

0:07:27.760,0:07:32.000
them might end up going down a dead 
end that won't carry on any further.

0:07:32.000,0:07:37.360
So I'll get a trunk. And I think the beanstalk 
thing is the fact that I've got this infinite

0:07:37.360,0:07:42.080
trunk that keeps going, but those other 
numbers that I could have tried from there,

0:07:42.080,0:07:46.160
the other options from here, those might 
be branches coming off it. Those don't

0:07:46.160,0:07:49.920
have to be infinite. I can have like 
finite little branches coming off.

0:07:49.920,0:07:55.200
So, does that work? Well, off zero. I tried one,

0:07:55.200,0:08:02.800
that didn't work. And I tried two -- actually, 
I'm going to do a bit of erasing again.

0:08:03.920,0:08:07.920
I'm going to start with zero much lower down 
now. Zero. And I know that that goes up to

0:08:07.920,0:08:14.480
three and then six, and then looking 
ahead of what I copied off the OEIS,

0:08:14.480,0:08:18.640
11, 15. Is that enough to get the idea? Let's see.

0:08:18.640,0:08:22.080
From zero, I could have tried one, 
but it didn't work and I could have

0:08:22.080,0:08:27.360
tried two. That didn't work. 
That's not a great two. Three.

0:08:27.920,0:08:37.520
So could four have worked from three? Let's 
see. So four is the sum of one square,

0:08:37.520,0:08:46.480
so four minus one equals three. So I could have 
got there from three. And the same with five. Six,

0:08:46.480,0:08:52.600
seven, ... where are we? Seven is the sum of four 
squares. So that comes off three as well, whoop!

0:08:52.600,0:08:55.200
I thought that might come from further up. So

0:08:55.200,0:08:58.960
then eight's the one that works 
-- sorry, six! And then eight.

0:08:58.960,0:09:06.240
Nine is, ... I need to write some 
more bits. You know what? Let's go

0:09:06.240,0:09:13.040
to the OEIS. Let's find this thing. There's 
a list of these. I want this table. So nine,

0:09:13.040,0:09:17.760
is the sum of one square. Well, I 
could have said that, couldn't I?

0:09:17.760,0:09:23.280
Stop there. Okay. So, nine. 
Oh, let's do these differences.

0:09:25.760,0:09:32.400
10 and 11 also come off eight. 12 comes 
off nine. So we've got a branch that's not

0:09:32.400,0:09:38.640
trivial here. We're going two steps along, 
but it will turn out to not work later on.

0:09:38.640,0:09:42.960
13, 14, 15 all come off 11. So...

0:09:42.960,0:09:49.680
The Beanstalk carries on upwards from 
15. Infinitely. So there we go, that's

0:09:50.480,0:09:54.240
the infinite least squares beanstalk. 
Glad I've made some sense of it.

0:09:54.240,0:09:57.200
Now there's loads of questions 
that this brings up that I would

0:09:57.840,0:10:01.200
I really like to answer from 
here. I'm not going to do it now.

0:10:01.200,0:10:04.720
How do we prove it's infinite? I 
think if we maybe know the fact

0:10:04.720,0:10:10.640
that this thing is always monotonic, 
maybe every... Well, ... here's an

0:10:10.640,0:10:15.840
easy question. Does every number fit 
on to this infinite beanstalk somehow?

0:10:15.840,0:10:18.720
The answer is yes, because, 
well sort of by induction.

0:10:18.720,0:10:26.480
If I've got some really big number, I then 
subtract the number of squares that you need.

0:10:26.480,0:10:30.640
This function, I need to give this a better 
name. The number of squares counting function.

0:10:30.640,0:10:35.360
Subtract that, I get a smaller number which, 
by induction, I've said is in the Beanstalk.

0:10:35.360,0:10:41.360
So I've covered a few things here. So 
I just need that to not be a gap of

0:10:41.360,0:10:48.320
four in this thing here. So, I don't know. 
Maybe that is enough, maybe I've got it,

0:10:48.320,0:10:50.880
maybe I need to do a little bit more thinking.

0:10:50.880,0:10:53.440
Lovely Beanstalk. Cool!