## Interesting sequences (Catalan, maybe?)

For the discussion of math. Duh.

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joshz
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### Interesting sequences (Catalan, maybe?)

For my math class, I have to find an "interesting" sequence to do a project on. It can't be too well-known, because my math teacher wants people to learn about new sequences. (so that probably means no Fibonacci, Collatz, etc.)

Does anyone know of any sequences that would be interesting for me to research?

Thanks!
Last edited by joshz on Fri Jan 30, 2009 12:30 am UTC, edited 1 time in total.
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

A_of_s_t
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### Re: Interesting sequences

Use an infinite series to find the area under a curve. That's always fun.

EDIT: Actually, that's called a Reimann sum and is well known IIRC.
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joshz
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### Re: Interesting sequences

Well, that's not really a sequence, just a way of analyzing sequences.
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

A_of_s_t
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### Re: Interesting sequences

My mistake, I got series and sequences mixed up -- a noob mistake.
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Cleverbeans
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### Re: Interesting sequences

Are there any restrictions on the sequences like they must be integers or real valued? Are you allowed to study a class of sequences or does it have to be a specific sequence?
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joshz
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### Re: Interesting sequences

Cleverbeans wrote:Are there any restrictions on the sequences like they must be integers or real valued?
Not AFAIK--just a sequence.
Are you allowed to study a class of sequences or does it have to be a specific sequence?
I *think* a specific sequence-it's a fairly small project. 1/2 a poster board and a 3-4 slide powerpoint.
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

rhino
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### Re: Interesting sequences

It's very well-known, but you could look at the sequence.
1, 1/4, 1/9, 1/16, 1/25, ...
And show that it sums to pi2/6

There is loads of history behind this problem so there's plenty to discuss that's of interest to a casual audience.
In my opinion, the nicest proof of this uses Fourier analysis, but there's a risk that this is either too simple (if you're at university) or too hard (if you're at school).
There is also a pretty but "informal" derivation of the result by Euler which is more elementary here
Wiki also has a formal proof of the result which is between the two arguments above in terms of the machinery required.

As an alternative, you could rattle off a proof that 1/2, 1/4, 1/8, 1/16, ... sums to 1
And then abuse the same proof to "show" that 1, 2, 4, 8, 16, 32, 64, ... "sums" to -1
Which could serve as a preamble to some discussion of p-adic numbers

Edit: typos
Last edited by rhino on Tue Jan 27, 2009 11:16 pm UTC, edited 1 time in total.

joshz
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### Re: Interesting sequences

I like it, but I think the math is too complicated (high school sophomore, advanced class), and there's not enough I can do with it--just that proof, which probably isn't that interesting to average people.
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

jestingrabbit
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### Re: Interesting sequences

You could do something with Grandi's series,

http://en.wikipedia.org/wiki/Grandi%27s_series

use it to illustrate that mathematicians can come at the same question from very different angles ie you can say that if s= 1-1+1-1+1... then 1-s=s, so s=1/2, but the sequence of partial sums isn't Cauchy, but it is Cesaro summable.

But its not very sequencey.
Last edited by jestingrabbit on Wed Jan 28, 2009 12:18 am UTC, edited 1 time in total.
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joshz
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### Re: Interesting sequences

Interesting, but I'm not sure I would be able to squeeze enough out of it for it to work well.
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You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

joshz
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### Re: Interesting sequences

Do you think Pascal's triangle would count as a sequence?
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

Buttons
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### Re: Interesting sequences

The Catalan numbers are probably a good choice.

If that's too simple, perhaps the Robbins numbers: 1, 1, 2, 7, 42, 429, 7436,.... These are known to count a few different families of objects (like alternating sign matrices or totally symmetric self-complementary plane partitions), but no bijections are known between these families. They're relatively new (~30 years old), and I'm sure your teacher hasn't heard of them. They also allow for lots of pretty pictures, especially the plane partitions and fully packed loop configurations.

In general, this is probably a good resource.

joshz
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### Re: Interesting sequences

I think you're thinking a bit too complicated. I'm a high school sophomore, remember.

I think I'll go with Pascal's triangle, and see if he likes it.
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

t0rajir0u
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### Re: Interesting sequences

Here are some sequences with fairly elementary definitions that are generally not studied at the high school level:

- The Bell numbers.
- Derangements.
- The Stirling numbers (although technically this is not a sequence).
- The Morse-Thue sequence.
- The Beatty sequence associated to the golden ratio [imath]\frac{1 + \sqrt{5}}{2}[/imath]. It has the pretty cool property that [imath]a_n[/imath] is the smallest number that hasn't appeared either as part of the sequence or as one of the values of [imath]a_k + k, k < n[/imath].
- The sequence of numbers whose base-3 expansion contains no 2s. It has the pretty cool property that no three values form an arithmetic sequence. (Write out a few terms in base 10; it's not at all obvious that this is the case.)

The Catalan numbers are a good suggestion, though.

Buttons
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### Re: Interesting sequences

Pascal's Triangle isn't really a "sequence", though. Plus, your teacher has certainly heard of it.

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### Re: Interesting sequences

If you just browse through the OEIS, you'll eventually come to something you can understand, but isn't too basic. I watched a 2.5 hour talk on the square triangular numbers, so there's plenty you could do with those. Continued fractions could also be used, but they're not a single sequence per se, but a method of representing every real number as a sequence of integers that can be truncated for an approximation of that number.
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joshz
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### Re: Interesting sequences

I kind of like the square triangular number idea, but are there any applications?
I *think* that may be kind of along the lines of what he meant by "interesting." (ie. fibonacci series shows up in nature)

btw, I agree with your sig, qinwamascot.
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

Ended
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### Re: Interesting sequences

Another idea: this sequence, i.e. xn+1 = cos(xn). (More precisely, the set of such sequences indexed by the starting value x0). It has an interesting tale of 'discovery' (Prof. Dottie, plus most people have probably done the 'hitting cos(x) over and over again' thing), and at least one nice property (converges to the same x for any x0). It might be interesting to look at the related sequences xn+1 = C*cos(xn), and investigate convergence for different values of {x0, C}.

Not many applications, though.
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joshz
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### Re: Interesting sequences

That's not really a sequence, though, is it? Isn't it just really the one number that's important in that?
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

Ended
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### Re: Interesting sequences

Yes, I suppose it's only really the limit which is interesting. But it's a perfectly valid sequence: xn = cos(n)(x0).
Generally I try to make myself do things I instinctively avoid, in case they are awesome.
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joshz
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### Re: Interesting sequences

Yeah, but what can I do with it/about it for a project?

Re: triangular square numbers: any applications? or any webpages on it that go into more detail than the first couple google hits?
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

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### Re: Interesting sequences

Buttons wrote:The Catalan numbers are probably a good choice.

Agreed!
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joshz
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### Re: Interesting sequences

Looking at those more, they seem pretty good (the Catalans). I at first didn't like them because the wiki about them was bad, but they seem pretty interesting.

Thanks everyone!

(Also, dèja vu for some reason. Odd.)
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

gmalivuk
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### Re: Interesting sequences

Yeah, while the Catalan numbers may be fairly commonly encountered in math in general, probably not so much in high school. (I first encountered them in college, for example, despite having taken a few college-level classes while still in high school.)
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### Re: Interesting sequences

I second t0rajir0u's nomination of derangements. The concepts are fairly easy to explain, and you can show the connection between the total number of permutations of n items (= n!) and number of derangements (= !n) and their relationship with e.

I also like his idea of Beatty sequences, and not just the ones for phi.* It's not too hard to show some interesting properties of these sequences using basic algebra & the definition of an irrational.

* But I will agree these are a pretty pair of sequences.

joshz
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### Re: Interesting sequences

Yeah, my teacher said the Catalans are too advanced.

Does anyone have anything else about triangular square numbers?
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

Buttons
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### Re: Interesting sequences

joshz wrote:Yeah, my teacher said the Catalans are too advanced.

Does anyone have anything else about triangular square numbers?

Seriously? I've taught fifth graders about the Catalans, but I can't imagine doing the same for triangular squares.

(On a side note, since when is it the job of the teacher to tell the student when something is too advanced for them? Especially when it isn't.)

joshz
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### Re: Interesting sequences

Maybe it was just the page I brought in, but he said I was "jumping into the deep end."
Also, really? These Catalan numbers?
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

t0rajir0u
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### Re: Interesting sequences

There are some pretty simple ways to understand the Catalan numbers: parenthesized expressions, binary trees, Dyck paths... these are all defined in a fairly elementary way and one can grasp immediately what these structures look like.

Understanding the triangular squares, on the other hand, requires a knowledge of the methods for solving Pell's equations, which is much harder. (It's not to say that there aren't hard things you can do with the Catalan numbers - they are very deep - but you can start simple and stay there if you like.)

joshz
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### Re: Interesting sequences

Do you have any links for that?
Because most of the stuff I can find is kinda complicated.

Otherwise, I'll probably need something else.

/me thinks I should be able to find something easier than this. My google-fu sucks.
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

gmalivuk
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### Re: Interesting sequences

I first encountered the sequence working with Dyck paths. Which you can totally talk about without having to name them that.

The MathWorld entry on Catalan numbers has a few graphical representations of other times they pop up.
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joshz
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### Re: Interesting sequences

I think that definitely gives me a place to start with that, without getting too complicated.

So far I have:
http://mathworld.wolfram.com/DyckPath.html
http://mathworld.wolfram.com/CatalanNumber.html
http://74.125.47.132/search?q=cache:6om ... ent=safari

Is there anything else pertinent I could look at about them?
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

samspotting
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### Re: Interesting sequences (Catalan, maybe?)

Generating functions from geometric sequence.

t0rajir0u
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### Re: Interesting sequences (Catalan, maybe?)

Describing the generating function of the Catalan numbers is highly nontrivial (for this audience), so that's probably a bad idea.

joshz
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### Re: Interesting sequences (Catalan, maybe?)

t0rajir0u wrote:Describing the generating function of the Catalan numbers is highly nontrivial (for this audience), so that's probably a bad idea.
What do you mean?

Makes perfect sense to me. And besides, I'll probably spend more time on applications than on the equation itself.
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

Buttons
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### Re: Interesting sequences (Catalan, maybe?)

That's a closed-form expression, not a generating function.

joshz
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### Re: Interesting sequences (Catalan, maybe?)

Okay?

OH! You were replying to samspotting. That makes more sense.
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

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### Re: Interesting sequences (Catalan, maybe?)

joshz wrote:
t0rajir0u wrote:Describing the generating function of the Catalan numbers is highly nontrivial (for this audience), so that's probably a bad idea.
What do you mean?

Makes perfect sense to me.

That's a closed-form expression, not a generating function, joshz.

joshz
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### Re: Interesting sequences (Catalan, maybe?)

Right, but why do I need a generating function?

(And also, I don't really know the difference, seeing as I'm still in high school.)
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.

t0rajir0u
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### Re: Interesting sequences (Catalan, maybe?)

The generating function itself has a simple closed-form expression: it is $\frac{1 - \sqrt{1 - 4x}}{2x}.$ Explaining why this function represents the Catalan numbers requires you to at least have some familiarity with calculus and ideally to have a strong understanding of how power series multiply. The fact that this function exists lets you do a lot of cool things with the Catalan numbers that aren't as easy to do otherwise, like prove certain identities. Again, the whole notion of a generating function is just not appropriate for this audience.