How many ways can you triangulate a regular polygon?
A triangle has one way, a square has two, a pentagon has five,
and a hexagon has 14.
How about for a septagon?
You can try this yourself, but has someone already figured
this out?
You could check this by looking up the sequence in the OEIS,
the Online Encyclopedia of Integer Sequences.
Searching for 1,2,5,14 produces over 350 sequences.
We find that our polygon problem matches the Catalan numbers
whose general formula is given here.
There are many other interpretations of this sequence:
it counts the number of paths in an n by n grid which
start in the bottom left corner, only move right and up,
and finish in the upper right corner.
The Catalan numbers also count the number of rooted binary trees with
n *internal* nodes where each node has zero or two children.
They count the number of ways for an even number of people
around a table to shake hands without crossing.
Lastly for now, the Catalan numbers count the number of
ways to order n pairs of open and closed parentheses.
The OEIS entry for this sequence also gives *formulas* involving
the Catalan numbers, such as sums, asymptotic growth, integrals,
generating functions, and determinants of Hankel matrices whose entries
are Catalan numbers.
There's plenty of other information as well.
This massive collection of integer sequences started over 50 years ago when
Neil Sloane, a young math student, was seeking information
about a particular sequence:
"Back in 1964 when I was working on my thesis at Cornell.
And I had a sequence, I had a lot of sequences I was trying
to identify, so I picked the simplest one to see if there
was a nice formula for that.
I was interested in how fast the formula grew.
So if I'd had a formula for the sequence, I could have
just looked at its asymptotic behavior and that was the
information that I wanted.
So there were various books around of combinatorics and other books of tables which
had sequences in them but I couldn't find the particular sequence
I was interested in there.
So I started writing on file cards, all the sequences that I did find
so my next sequence I would have a place to look.
So I started making a little catalogue of sequences.
And it grew and it grew."
And grow it did.
In 1973 Sloane published his first catalogue containing over 2000 sequences.
The second book with over 5000 sequences appeared in 1995.
The next year the OEIS web-site was launched with 10,000 sequences.
As of 2015, over 250,000 sequences are in the OEIS.
The site is maintained by a team of 29 editors-in-chief
and over 100 associate editors.
There are so many fascinating sequences.
Some are well-known, like sequence A5 which gives the the number of divisors of
n.
Interesting but less known is sequence A48987, the number
of possible chess games after n moves from each side,
or sequence A19473, the number of n-celled “still lifes”
in Conway's Game of Life.
And does Sloane have favorites?
"My favorite sequences --- well, you know over
the years that's changed a bit.
My all-time favorite is Recaman --- Bernard Santos Recaman's sequence,
A5132, which is just a wonderful sequence.
In fact, it is my favorite one to listen to."
"My favorite sequences --- well, you know over
the years that's changed a bit.
My all-time favorite is Recaman --- Bernardo Santos Recaman's sequence,
A5132, which is just a wonderful sequence.
In fact, it is my favorite one to listen to."
In fact, we've been listening to Recaman's sequence during this entire video!
Each sequence has its own music.
Now a question for you.
It concerns the EKG sequence A64413.
>From the description, can you figure out the next number?
Give it shot, OR, look it up at oeis.org