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The On-Line Encyclopedia of Integer Sequences
Neil Sloane writes, "Run across a number sequence you want to identify?
For instance, what comes next after 1, 2, 4, 9, 20, 48, 115, 286, 719, ...?
The
On-Line Encyclopedia of Integer Sequences is a database
with over 50,000 such sequences. Serves as a "fingerprint file,"
so you can see if your problem has been studied before.
Widely used by researchers in number theory, combinatorics,
computer science, physics, chemistry, etc., as well as people
trying to solve puzzles. " That's nuts. Mind you it would in no way have assisted me in getting a decent grade in calculus, but still, it's fun.
Genetic sequence indicating the probability of an NT server crashing under various loads.
Zilog had the Z8, Z80, Z800, Z8000 and Z80000. Then the company suffered a fatal arithmetic overflow trap.
Mea navis aericumbens anguillis abundat
Maybe not for calculus, but this facility is very useful when you study discrete structures. I've used it many times, found several connections between seemingly unrelated structures, and sometimes had to feel embarrassed for not seeing the obvious pattern myself.
In a similar vein, and very interesting for coding theorists, is this page. Set up by kernel and nethack hacker Andries Brouwer.
Most intelligence tests have sections where the test taker tries to figure out the next number/letter in a sequence. I wonder if a person were to study the sequences in this database, if they would be able to 'raise' their IQ as measured on standard tests, or if exposure to the different types of sequences doesn't correlate well to the ability to figure out a given sequence.
...
Of course, as another poster stated, any given finite sequence has an infinite number of polynomials that can generate it and any other term you choose, which is why those types of questions tend to irritate me. The question should be qualified, as in, 'What is the next number in this sequence, assuming a simple generator for the sequence?' (Leaving room to quibble over the meaning of the word 'simple', naturally)
Definitely a very cool site, and I am glad to see this type of stuff here.
Since we are on the topic of sequences, and there was another article about puzzles, here's an old chestnut:
What is the next letter in the following sequence?
O T T F F S S
Its full of cool sequences. All the major random number polynomials are there. Any standard useful polynomial that I know about is there. I wonder how many of the poor crypto systems have their core sequences in there already. I know where I'm going the next time I've got a few numbers I don't understand. I just wish it had the floating point sequences for things like taylor series factors for cosine.
It's true that you can concoct a polynomial which will spew out any finite sequence of numbers, and that this means you can't *guarantee* you've got the formula for a sequence just by checking finitely many terms.
However, if you find a formula that "seems right", it may make it easier to prove that it *is* right, because now you're barking up the right tree.
An example of this happening in real life is the number 196884. It turned up in two seemingly unrelated places, in the character table of the Monster Group and in the expansion of the j function. This lead mathematicians to search for - and find - the connection between the two.
See Scientific American for a good article about this "Moonshine Conjecture".
perl -e 'fork||print for split//,"hahahaha"'
It was a joke. Barney on the Simpsons was the one doing "Number four."
One of the great ways this is useful in that it provides pointers to research papers. It keeps people from reinventing the wheel regarding the sequence, by giving a lot of information on what has already been done.
6, 14, 23, 28, 33, ?
I mean translate those numbers to pitch and duration and you have an instant hit...
Je t'aime Stéphanie
for any 'n' points, there is a (unique) polynomial of degree 'n-1' (or less) that takes these values for x = '1,2,3,4,5,6,....,' and an infinite number of higher degree polynomials. So generating a finite number of points, doesn't guarantee you've detected which sequence you're generating.
Athletic Scholarships to universities make as much sense as academic scholarships to sports teams.
Yes, Ian Witten and Lloyd Smith at the University of Waikato built one where you can whistle, hum or sing into a Java applet and it would find matching themes. ISTR that it's actually the rhythm that most strongly identifies the theme - everyone can tell that dit-dit-dit-dah is beethoven's 5th Symphony, even stripped of all pitch info. Add contour, andf you're away. It's called Meldoex - Melody index.
p =coltitle for a demo.
The paper they wrote is Smith, Lloyd A., Rodger J. McNab and Ian H. Witten. Sequence-based melodic comparison: a dynamic-programming approach. In Hewlett, Walter B. and Eleanor Selfridge-Field (eds.) Melodic Similarity: Concepts, Procedures, and Applications, Computing in Musicology 11, Chapter 4, 1998, p 101--117.
Check out http://www.nzdl.org/cgi-bin/gwmm?c=meldex&a=page&
Ah, fun with Pythagorian triplets.
:)
I know of two "generators" for triplets, but I don't think it is helpful for the x^2, (x+1)^2 series (except in the very basic case of 3-4-5).
Anyway, for all natural numbers n:
If n is odd, then n, floor(n^2/2), ceil(n^2/2) is a triplet.
If n is even, then n, (n/2)^2-1, (n/2)^2+1 is a triplet.
A little algebra will show why these are true, but it is interesting how it starts by catching some of the better known triplets.
(3-4-5, 5-12-13, 7-24-25, 8-15-17, etc.)
Now if only the site becomes un/.ed, I might not get any work done today.
----
My UID is the product of 2 primes.
I tried giving the Encyclopedia the ISO-RR33 benchmark integer sequence 99 bottles of beer on the wall..., but it failed to even parse the request. So I simplified it to the integer values in the first six-pack: 99, 98, 97, 96, 95, 94. This time it parsed the request, but said the sequence wasn't in its database! What good is this site if it doesn't event recognize the beer sequence?
--Jim
I've had occasion to use this and thought it was pretty cool. There have been printed versions of these, but the online one is better.
:= 1,2,1,1,1,1,1,1.....
Another interesting idea that I've seen printed is a musical theme dictionary, if you can plunk out the first few notes by ear then you can look up the sequence. Has anyone done this online? Would someone sue you for it, since printed and/or recorded music is a pretty touchy subject on the Internet.
My favourite sequence, not listed, is:
s(n)
n=1,2,3,4,... is the number of people in an elevator and, if one of them farts, s(n) is the number of people who are sure who did it.
Alan.
Wow this has been a subject that nobody seems to want to say anything about.
;-)
I just wanted to say thanks to Rob for running this one though - I found the significance of a very interesting series which is related to the solution to:
x^2 + (x+1)^2 = z^2 (x,z in natural numbers)
That series is: 1,3,7,17,41,99,239,577,1393,3363,...
Each subsequent number in the series converges on a multiple of the previous one, but according to the site the series is also the numerators in the continued fraction expansion of the square root of two.
(Score -1: Boring)
I'm not a journalist, but I play one on slashdot
ID Number: A0348265 8,85849,226980,601373,1594870,4232100,11 230771,29798539,79034638,209526631,555172356,14701 95001,3891131705,10292857772
Sequence: 0,1,1,2,4,9,20,48,115,286,719,1841,4755,12410,325
Name: n-node rooted trees of height at most 9.
Links: Index entries for sequences related to rooted trees Transforms
Formula: Take Euler transform of A034825 and shift right. (Christian G. Bower (bowerc@usa.net)).
See also: See A001383 for details.
Keywords: nonn
Offset: 0
Author(s): njas
It only hurts when you survive
Another great resource is the Inverse Symbolic Calculator. Take that real number you've been trying to identify, and see what formula or combination of known constants might have generated it.
The integer sequence database has proven quite handy to me on several occasions. Kudos to N. J. A. Sloane for creating and maintaining it, and to the people who keep contributing more good sequences!
-jason
"If you're not part of the solution, you're part of the precipitate."
Encyclopedia of Integer Sequences by N.J. Sloane and S. Plouffe, USD$57. It is actually neat; I found it in a (university) library once. There is a fine line between "combinatorics" and "recreational mathematics" sometimes, and that's good. The book will certainly have a large number of sequences that you'll find interesting if you have any interest in mathematics whatsoever. Other sequences are horribly technical. It's a very useful book and not as boring as some of the previous posters think.
--- Premature complacency is the evil of all roots