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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.
0, 1, 2, 720!, ?
Bruce
Finally, we have a certifiable, 100%, no-holds-barred, dyed in the wool, true blue, hi-fi, back-in-the-day, old school "News for Nerds" story, and this is your response?
Go back to your ESPN, frat boy. Crack another beer while you're at it.
grits in bowl, open pants, tilt bowl, pour grits into pants. thank you.
I remember when I was in elementary school and high school, teachers would sometimes pose difficult problems that would stump the class. Maybe one kid would figure one out. With resources like these available, those days are going the way of the dodo.
There is a fine line between "combinatorics" and "recreational mathematics" sometimes, and that's good Indeed. But sometimes I think that they are all for masochists: the problem statements are often easy and the solutions are often fiendishly difficult. In contrast, many other fields of mathematics are hard to learn but have lots of interesting problems that can be reasonably solved once you have the background. Maybe that's just the perspective of a non-combinatorial mathematician.
Just this morning a student and I were investigating an enumeration problem in lattice paths. After we generated a few base-case counts by hand, I tried showing him how to use Sloane's wonderful online reference. "How strange" I replied, when seeing the server unable to handle my humble request. Thank goodness the site was "only" slashdotted!
Actually that was sequence 81, one of the most basic:7 3,87811,235381,, 268282855,743724984,1 86554308,354426847597
%I A000081 M1180 N0454
%S A000081 1,1,1,2,4,9,20,48,115,286,719,1842,4766,12486,329
%T A000081 634847,1721159,4688676,12826228,35221832,97055181
%U A000081 2067174645,5759636510,16083734329,45007066269,126
%N A000081 Rooted trees with n nodes (or connected functions with a fixed point).
%D A000081 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
%D A000081 F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 232.
%D A000081 N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 42, 49.
%D A000081 D. E. Knuth, Fundamental Algorithms, 3d Ed. 1997, pp. 386-88.
%D A000081 F. Bergeron et al., Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 279.
%D A000081 R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
%H A000081 Index entries for sequences related to rooted trees
%H A000081 Index entries for sequences related to trees
%H A000081 Index entries for "core" sequences
%H A000081 ECS 57
%F A000081 G.f. A(x) = x exp(A(x)+A(x^2)/2+A(x^3)/3+A(x^4)/4+...)
%F A000081 Also A(x) = Sum_{n>=1} a(n)*x^n = x / Product_{n>=1} (1-x^n)^a(n).
%F A000081 Recurrence: a(n+1) = (1/n) * sum_{k=1..n} ( sum_{d|k} d*a(d) ) * a(n-k+1).
etc.
Neil Sloane (njas@research.att.com)
That's sequence A000396 in the database, one of the oldest5 3842176,6 9216
(1 is not normally regarded as a perfect number)
Here is the beginning of that entry:
%S A000396 6,28,496,8128,33550336,8589869056,137438691328,
%T A000396 2305843008139952128,26584559915698317446546926159
%U A000396 1915619426082361072947933780843036381309973215481
%N A000396 Perfect numbers.
%D A000396 Uhler, Horace S.; On the 16th and 17th perfect numbers. Scripta Math. 19 (1953), 128-131.
%D A000396 B. Tuckerman, The 24th Mersenne prime, Notices Amer. Math. Soc., 18 (June, 1971), Abstract 684-A15, p. 608.
%D A000396 B. Tuckerman, The 24th Mersenne prime, Proc. Nat. Acad. Sci. USA, 68 (1971), 2319-2320.
%D A000396 J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
%D A000396 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 19.
%H A000396 Cunningham project
%H A000396 Perfect numbers
%H A000396 J. S. McCranie, A study of hyperperfect numbers, J. Int. Seqs. Vol 3 (2000) #P00.1.3
%F A000396 The numbers 2^(p-1)(2^p - 1) are perfect, where p is a prime such that 2^p - 1 is also prime (see A000043, the Mersenne primes), and it is believed that there are no other perfect numbers.
Extending this sequence is equivalent to computing Mersenne primes.
Neil Sloane (njas@research.att.com)
I scoff at your whimpy sequence mortal!
ID Number: A052200Sequence: 1,64,20736,16777216,25600000000,63403380965376, 232218265089212416,1180591620717411303424, 7958661109946400884391936,68719476736000000000000
Name: Number of n-state 2-symbol 5-tuple Turing machines.
References J. P. Jones, Recursive undecidability - an exposition, Amer. Math. Monthly, 81 (1974), 724-738.
Formula: (4*(n+1))^(2*n)
See also: Cf. A028444, A004147.
Keywords: nonn,easy,huge
Offset: 0
Author(s): Michael Somos
(somos@grail.cba.csuohio.edu) - 28 Jan 2000
Now that's a sequence. RAAAGH!
Actually, part of the reason might be that Slashdot was experienceing server problems. I couldn't get through to /. for about an hour, and then when I first got through, I got a server error.
That because it is, bar none, the most booooring stoy I have EVER read. A dictionary of number sequences? Ye Gods!
I did a report on him and sphere packing in 10th grade!
The Lexington Avenue line doesn't stop at 79th street.
Also, there is no 6th St. stop on any line.
Because I didn't know about. Nw I do, thank you.
Spencer Ogden
Back in high school (mid 80s) we had very informal my-dick-is-bigger-than-yours type contests to see who could pointlessly waste the most amount of CPU time on the timeshared PDP-11 within a marking period. Every marking period, the system administrator would print out a list of the active accounts on the system, which showed total amount of time logged in, total amount of CPU time used, etc. If your account had the most CPU time, it was a (dubious) distinction. It meant you were working hard, I guess.
So having compute-bound programs running all the time generating sequences like this one was a briefly popular craze.
Anyway, 1,6,28,496,8128 is as far as this sequence goes in 16-bit integers. But some of you out there might actually have 32-bit or 64-bit computers by now, so maybe this can be extended, if you know what the sequence is. Hint: it has to do with factors.
I bet this one's not in there.
(Actually that's a link to a pretty cool random number server. They also have links to some others based on radioactive decay, etc.)
---- "If we have to go on with these damned quantum jumps, then I'm sorry that I ever got involved" - Erwin Schrodinger
... E N T E ...
which also happens to be the german word for duck, believe it or not...
Pentium. Additionally, 80186 should really be 8088 in that list, I believe. I think there was an 80186, but it wasn't an intel chip - it was a spruced up 8088 clone that came out AFTER the 80286.
Either that, or I've totally scrambled my cache...
-- Abigail
I tried looking up "1 2 3 4 5 6" in the encyclopedia, but it couldn't come up with any answers (I asked it for the top 30). So either their server is very incomplete, it's been slashdotted, or it was trying to calculate all possible answers before giving me back the top 30.
Were you joking, or did you mean "Number nine. Number nine...."?
Weak Monty Python reference...
If you think about it for a few, you will find
a cute little algorythm for extracting the
square root of a number.
These are the first few perfect numbers, i.e. numbers whose proper factors (including 1) sum to the original number.
perl -e 'fork||print for split//,"hahahaha"'
E, of course. Followed by N and T.
-Bob
E. Actually, ENT. Like Ear-Nose-Throat Doctor.
4004,8008,8085,8086,80186,80286,80386,80486 ... ?
> Inverse Symbolic Calculator
> http://www.cecm.sfu.ca/projects/ISC/
Argh. NOW I found out about a home site, after I graduate. *sigh*
Another great site is Eric's Treasure Trove of Science
http://www.treasure-troves.com/
Cheers
I agree with that.
/pr
--
Number four. Number four. Number four. Number four. Number four. Number four.
is mathworld.wolfram.com I know this is slightly offtopic, but mathworld references Sloane all the time so the association came to my mind. mathworld is a great fully cross-referenced moderately in-depth encyclopedia on everything math (by Eric Wiesstein, I think). Iy's an awesome resource--if you like math, check it out.
-- The Sheep --
oops
POSTED: 11:52pm (TZ?)
FIRST POST: 10:38am EST
./ server responded to ping requests okay but I've been getting request time outs all day
Can anyone say DDoS?
take a triptonica to subthunk
It is interesting to note that these same number theories "recreational mathematics" have helped us explain many theories and puzzles in the physical world. For example matrices were simply a mathmatical mind game when first devised and no actual use for them was imagined. Of course later it was realized that they could be aptly used in all sorts of real life scenarios such as vector algebra and robotics "inverse kinematics".
Actually most branches of mathematics no matter how abstract have eventually found their way into physics or some other related scientific field. Does this not strike you as somewhat strange? I mean we devise up these crazy number theories and ideas and then they almost magically appear in nature to fit our mathmatical model.
Just some food for thought...
Nathaniel P. Wilkerson
NPS Internet Solutions, LLC
www.npsis.com
Nathaniel P. Wilkerson
www.haidacarver.com
It was "Number eight" you fool!!!
Je t'aime Stéphanie
Oh where oh where has the integer book beeeen... I could've used some help with my discrete mathmatics and theroy courses last semester. Oh well, I beleive I'll be purchasing this book soon anyways... :-)
------- What exactly is real?
Would this be better if you could enter simple equations into the database or fractions? So it can accept infinate series? Would be greatly helpfull to Calclus students.
"The anwser to the ultimate question of Life, the Universe, and Everything is... 42" -Douglas Addams
So wait a minute, /. slashdot-effected itself? Strange, but I didn't see any articles referencing www.slashdot.com. If it did, I dunno what the effect would be. Server would be down for weeks, probably.
:)
Speeding never killed anyone. Stopping did.
I think you mean slashdot.ORG putz!
Speeding never killed anyone. Stopping did.
No, but it certainly gives you a good head's up. If you find that the sequence you've got by method A matches the sequence someone else got by method B, it certainly indicates that you should spend some effort looking for a link.
14,23,28,33,42,51,59,68,77,79,86 Thats a list of NYC 4/5/6 subway stations ... Easy -- tttfn -reub
-- ttfn -reub "The best is the enemy of the good." - Voltaire
Genetic sequence indicating the probability of an NT server crashing under various loads.
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