Last updated Jun 07 2008. I need to update this page to refer to the new OEIS at oeis.org - N. J. A. Sloane, Jan 15 2011.
There is already a lot of information about
the Encyclopedia on other pages,
and some of these answers
will just be pointers to the appropriate page.
Q: What is the purpose of the OEIS?
A: The main purpose is to allow mathematicians
or other scientists to find out if some sequence
that turns up in their research has ever been seen before.
If it has,
they may find that the problem they're working on has already been solved,
or partially solved, by someone else.
Or they may find that the sequence
showed up in some other situation,
which may show them an unexpected
relationship between their problem and something else.
Another purpose is to have an easily accessible database of important,
but difficult to compute, sequences.
For example, if you're testing
some conjecture about Mersenne primes, you can look up the ones that are
known (see A000043),
rather than spending years recomputing them.
For more information on this point, see the Demo files and the Welcome page.
Q: How do I cite the OEIS in a paper?
A: Click
here.
That is a link to a section of the
Welcome page,
which has a lot more information about the database.
Q: Can you give some examples of successful
applications of the OEIS?
A: See the
list of papers
that have acknowledged help from the database;
also the
comments from readers
on the last page of the
Demo
files.
Q: Some advice for new users and new contributors?
A: (From Jonathan Post (jvospost3(AT)gmail.com), Jun 07 2008)
New OEIS contributors should be strongly encouraged to use the OEIS
webcam to see the breadth and variety of those seqs deemed "nice." It
is the equivalent of listening to a radio station that plays the
greatest classical or jazz or rock songs (and for that auditory matter
using the "listen" feature), or walking through a museum of unusually
beautiful Mathematics (and, for that visual matter, encouraging the
use of the "graph" feature).
Another positive message to new OEIS contributors, to enhance that sense of community, and to provide a balance against "making stuff up" without reference to what is known and had been judged interesting, is, beyond looking for duplicates, looking from triples and n-tuples of sequences that are implicitly related, and making that relationship explicit, perhaps by showing that these are different rows or columns or diagonals of the same previously unshown array. Or by making the analogy: sequence A is to sequence B as sequence C is to sequence D.
Or by using the transforms available on some sequence previously not so transformed.
The goal is, not to merely reward ("nice") or punish ("less" or "probation" or silent deletion) those contributors externally, but by enhancing their intrinsic motivation to make a contribution more likely to be appreciated.
Q: OEIS Summer Rules, Jun 02 2008, but still valid today!
A:
Thank you!
Neil Sloane
Q: How many terms do I need to look up a sequence?
A: It depends!
Usually you should enter about 6 terms, starting with the second term.
Leave off the first term or two, because people may disagree about where
the sequence begins.
Don't enter too many terms, because you may have more terms than are in the OEIS
right now!
But one regular user says:
Don't hesitate to try the OEIS even if you only have a few terms.For more information about looking up sequences, see the hints file.
For example, the two terms 2, 1729 identify the "taxicab numbers" A011541.
For a better example, the single number 15170835645 identifies uniquely A003825.
Q: How do I find which sequences cross-reference
a particular sequence?
A:
Look up the A-number as a word (rather
than as a sequence number).
For example, to find all the sequences that mention
the Narayana-Zidek-Capell numbers
A002083,
enter A002083 into the search window.
If you only want to see sequence A002083, enter id:A002083 .
You could also enter Narayana Zidek into the search window.
For more information about looking up sequences, see the hints file.
Q: I looked up a really basic sequence (the number
of abelian groups of order n, in fact),
and I was surprised to find that it wasn't there. How come?
A: You probably miscalculated one
of the terms! It is there: A000688.
After 35 years most of the
basic sequences are in the database now.
When this happens, please recheck your calculations
before submitting the sequence.
It is also possible that you have included
an initial term that most people don't include
(perhaps starting a number-theoretic
sequence at n=0 rather than n=1).
Of course it is entirely possible that your sequence really isn't in the database, in which case please submit it!
Q: I've heard about the Motzkin numbers,
but I don't know the beginning of the sequence,
so how can I find them?
A: (1) Enter
Motzkin numbers
into the
lookup page. Or
(2) use the Index
to the OEIS.
Incidentally, the Motzkin numbers are A001006.
Q: I was trying to find the entry for the Fibonacci numbers,
but when I entered 2, 3, 5, 8, 13 I got too many matching sequences.
A:
(January 2006: this problem should now be fixed. The
following is the old answer to this question, which is still helpful.)
Use the Index to the OEIS.
If you are in the right ballpark, the replies will be sorted with
the "core" sequences first, followed
by the "nice" sequences,
and your sequence should be the first one that is listed.
But for common beginnings, like 2, 3, 5, 7 or 2, 3, 5, 8
there are a lot of possible continuations,
and you should give more terms if you have them.
Otherwise, try the
Index.
Or go to the
Welcome page,
and download the section of the database
containing sequences around the one you are
interested in
(in the lexicographic order),
and then you can browse. But beware, those files
are all quite large.
Incidentally, the Fibonacci numbers are A000045.
Q: How do I find all the sequences that mention my name?
A:
Enter your name in the search page.
To find just those where your name is on the Author line,
enter author:Smith for example.
To find just those where your name is on the Extension line,
enter extension:Smith for example.
Q: How can I get hold of all the sequences submitted by
John Smith that mention "lattice", so I can analyze
them on my computer?
A: Simply enter
author:Smith lattice in the search page.
Q: Sometimes a sequence will say
"n^2 + 1 is prime",
sometimes it says
"Primes of the form n^2 + 1".
What's the difference?
A:
A description like
"n^2 + 1 is prime"
means that the sequence gives a list of the values of n
such that n^2 + 1 is prime,
that is, 1, 2, 4, 6, 10, 14, ... (A005574).
On the other hand, "Primes of the form n^2 + 1" means that the sequence gives a list of the actual primes, 2, 5, 17, 37, 101, 197, 257, ... (A002496).
It is easy to get them confused! (But in this case it is easy to tell the difference: 4 is not a prime.)
Q: Where can I find an explanation of all the different keywords,
like "core" and "nice"?
A: In the page that describes the
format used
in replies from the database.
Note that the keywords "huge" and "done" are no longer used.
Q: What does the "offset" mean?
A: It tells you the subscript of the
first term in the sequence.
Examples:
In the internal format for a sequence the offset line (the %O line)
contains two numbers.
The first is the offset as just defined.
The second
gives the position of first entry greater than or equal to 2 in magnitude
in the sequence
(or 1 if no such entry exists), starting counting at 1.
The second offset is used to determine the position of the sequence
in the lexicographic order in the database.
For further examples, see the "offset" section of the internal format page.
Q: What does the keyword "more" mean?
A: This is a sequence where more terms are needed.
Ideally the entry gives enough terms
to fill three lines on your screen, like this:
%S A027614 1,1,3,14,80,468,2268,10224,313632,9849600,21954240,8894136960, %T A027614 105857556480,20609598562560,650835095904000,80028503341516800, %U A027614 5018759207362252800,503681435808239001600,56090762228110443724800or
%S A000004 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, %T A000004 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, %U A000004 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0If you can extend a sequence which has fewer terms than that, please do so, even if the keyword "more" is missing.
See also
What is the format to use when sending in more terms?
You can set the
WebCam
to browse the sequences that need extending,
or use the
main look-up page
to search for keyword:more.
Q: What does the keyword "base" mean?
A: A sequence where the definition
depends on which base we are using.
Palindromes (numbers
which are unchanged if the order of the digits is reversed, A002113)
are a classic example.
Q: Can you give an example of a "dumb" sequence?
A: The number of pages in the n-th volume
of the Harry Potter series of children's books.
This was actually submitted! It was rejected for many reasons,
one of which is that it is not well-defined
(English edition
or American? Hard-cover or paperback?).
Q: Why are there sequences with keyword "dead"?
A: Usually they indicate a published
sequence that was wrong.
The point is that if someone sees the erroneous sequence in a book
and looks it up in the OEIS,
the entry will point them to the right sequence.
Q: Which sequences should I submit to the OEIS?
A: Any that actually show up in your research.
If you looked up a sequence in the OEIS and were disappointed that it wasn't there, you should probably submit it so the next person who looks for it will find it.
Also, most sequences that appear in published papers should be in the OEIS. That way, if someone sees a sequence in a paper and looks it up, they'll be able to find out if anything new has been learned about the sequence since the paper was published.
Even if a published sequence happens to be wrong, it should be in the OEIS, with a cross-reference to the correct sequence.
Q: If I just make up a definition of a sequence,
should I submit it?
A: Probably not.
It's very easy to define new sequences, but most of them
are unlikely to be useful to anyone.
There are exceptions. If it is a really beautiful sequence, go ahead and submit it! See the EKG sequence (A064413) for an example of a made-up sequence of great interest.
Q: What should I do
before submitting a new sequence?
A: Check for connections with
sequences already in the database.
For example,
plot the sequence (or its log) to check for clues that it could be
formed by intertwining two known sequences.
For triangles or tables, check the columns, row sums, skew diagonals etc. to see if they are already in the database. If so, mention this in the submission.
Check your sequence with Superseeker
Study the first differences of your sequence to see if they provide any clues.
Make sure you give the correct offset (see above)!
If there is a relation to a geometric or combinatorial problem, describe this: it's valuable information.
And finally, if possible, please give a formula!
Q: How long should I spend on a submission to the OEIS?
A: After you've computed enough terms to fill 3 lines,
or as many terms as
you can, you should probably spend at least an hour writing and checking
the definition, comments, references, cross-references, etc.
If you don't think the sequence is worth spending an hour of your time on, then it's probably not worth having in the OEIS. Take the time to make sure that everything is accurate and explained clearly, so that someone who hasn't seen the sequence before will understand how it's defined and why it's interesting or important. Include any references that you know about.
Remember the advice given in the submit new sequence page:
IMPORTANT: Thousands of people use the sequence database every day. Please take great care that the terms you send are absolutely correct. The standards are those of a mathematics reference work.
Q: What "Subject" line should I use when
submitting a new sequence or comment?
A: If you are sending these by email,
rather than using the
submit new sequence or comment
web page, please use:
Subject: NEW SEQ for unnumbered new sequences,
Subject: PRE-NUMBERED for pre-numbered new sequences,
Subject: COMMENT for comments on existing sequences,
Subject: EDITED for a sequence that you have completely edited.
It makes the editing process easier if there is only one type of comment in a message. That is, please don't mix "Comments" and "New Sequences" (or "Edited Sequences") in a single email!
Q: I am sending in a comment on one of the sequences.
Should I send a copy to the
SeqFan mailing list?
A: No, that's not necessary.
Q: What kinds of number may appear in a sequence? Can a sequence include numbers that begin with 0?
A: No! (except for 0 itself)
Numbers in sequences must be positive or negative integers (or 0).
Acceptable numbers are 1, 5, 101, 0, -25, etc.
Unacceptable numbers are: 020, 7.125, 0.725E03, 3/4, 22?, 123...6 6^6^6^6.
But see also the section in the hints file on dealing with fractions and real numbers.
Q: A comma appeared in the middle of a number in my sequence - why?
A: Because you split a number across two lines!
My formatting programs assume numbers in a submission
are separated by commas, spaces or newlines.
If you mouse a sequence into the window, the mouse may introduce
newlines, like this:
8,12,20,24,32,36,48,52,60,68,80,84,96,100,112,120,128,140,144,152,168,172,19 2,200...That would cause the number 192 to appear as 19,2
Q: Should one specify links to the
Index when submitting sequences?
A: You certainly may if you know
what you are doing! There are plenty of examples in the database.
Just include the link in a links window on
the submission page.
Here is a typical link, to the index entries dealing
with the Goldbach conjecture.
But when you type it, please type "greater than", "less than", and double-quotes as single characters.
Don't type "less than" as "ampersand l t semicolon" as I had to
to make this line visible!
<a href="Sindx_Go.html#Goldbach">Index entries</a> for sequences related to Goldbach conjecture</a>.
At some later time you should then send me a list of updates to the index itself.
Q: What are some of the reasons why sequences are rejected?
A: Common reasons are:
Example: primes that contain the digits 2003. The "2003" is an arbitrary and large parameter.
Example:
numbers n such that the digits of n appear in all powers n^s with s = 1 through 20:
0, 1, 5, 6, 10, 25, 50, 60, 76, 100, 101, ...
The parameter 20 here is arbitrary. If we replaced it by 38, say, the sequence
would change.
Example:
digits of n appear in n^2, n^3 and prime(n):
976, 5903, 10513, 68793, 94682, ...
Too contrived!
Example: even numbers that are not the sum of two primes:
2, ...
Too short! Assuming the Goldbach conjecture is true, there are no more terms.
On the other hand, there are several legitimate sequences
based on the conjecture, e.g.
A002372. See also the
Index entries for sequences related to Goldbach conjecture.
Example: superfactorials,
1, 4, 6^6^6^6^6^6, 24^24^...^24 (with 24 copies of 24), ...
- contains unacceptable numbers.
On the other hand, number sequences that have actually appeared on quizzes or tests are welcomed. One of the reasons for the OEIS's existence is to help people with such tests!
Q: How can I find out if the
sequence I submitted was accepted or rejected?
A:
Q: What notation should I use in equations in
the comment and formula lines?
A: Try to use notation that's as close
to standard mathematical notation as is possible using ASCII text.
Don't use notation that's specific to Maple, Mathematica, PARI, or some other computer language, except in %p, %t, or %o lines. In particular, the arguments of functions should be enclosed in parentheses, not brackets; most function names should not be capitalized. Here are some common examples:
Use: | to mean: |
---|---|
binomial(n,k) | the binomial coefficient n-choose-k |
sigma(n) | the sum of the divisors of n, A000203(n) |
phi(n) | the Euler phi (or totient) function of n, A000010(n) |
mu(n) | the Moebius function of n, A008683(n) |
pi(n) | the number of primes <= n, A000720(n) |
prime(n) | the n-th prime, A000040(n) |
omega(n) | the number of distinct prime divisors of n, A001221(n) |
bigomega(n) | the number of prime divisors of n, with multiplicity, A001222(n) |
sqrt(x) | the (positive) square root of x |
floor(x) | the largest integer <= x |
ceiling(x) | the smallest integer >= x |
round(x) | the integer closest to x. |
In most cases, it's a good idea to explain what function you mean, since some of these function names aren't standardized, or are used for other things. For example, "pi(n+1)" might also mean the number pi multiplied by n+1. Use cross-references to other sequences to help with the explanations.
Q: Is it OK to use LaTex or Maple notation in equations?
A: No, please don't!
Use notation that can be understood by humans.
Say (1+x)/(1-x) not $\frac{(1+x)}{(1-x)}$.
Say n^2/2 not 1/2n^2.
Say A/(B*C*D) not A/B/C/D
Say sigma not $\sigma$.
Q: Should I use juxtaposition to denote multiplication, or should I put
a "*" or "." between the things being multiplied?
A: Either juxtaposition or an asterisk (*) are OK.
Don't use a dot to denote multiplication.
Q: How should I indicate summations?
A: There are many acceptable ways to do this.
All of the following examples are OK.
sum_{i=1..n} i^2 + i
sum_i=1..n (i^2+i)
sum_{d|n} d^3
sum_{ 2 <= p <= n, p prime} p^2
Other styles also acceptable, as long as they are clear!
Q: Should I denote infinity by "oo"?
A: Generally it is better to say "infinity"
explicitly.
Q: What does "lgdegf"
mean in a reply from Superseeker?
A: It stands for "logarithmic derivative",
which is nothing more than the derivative of the log of a function:
d log(f(x)) f'(x) lgdegf f(x) = ----------- = ----- dx f(x)
The replies from Superseeker are sometimes hard to read, I admit!
For example, suppose it says:
SUGGESTION: LISTTOALGEQ FOUND ONE OR MORE ALGEBRAIC EQUATIONS SATISFIED BY THE GEN. FN. WARNING: THESE MAY BE ONLY APPROXIMATIONS! Equation(s) and type(s) are: 2 3 [8 - 12 a(n) + 6 a(n) - a(n) , lgdegf]What this means is the following.
2 3 8 - 12 a(x) + 6 a(x) - a(x) = 0So you solve that for a(x), which is f'(x)/f(x), and then try to solve for f(x).
Like I said, this takes a bit of getting used to! But it can be very helpful.
Q: What is the best format to use when sending in
more terms for an existing sequence?
A: Please put the full sequence (all the terms,
not just the new ones) in the first window of the
Contribute new seq. or comment
web page.
In the "Comments" window, put something like "More terms". Or if you have found a mistake, say "Corrected and extended".
If it is a signed sequence (that is, contains negative numbers), just give the signed version in the top window, and don't bother to include the sequence of absolute values. The updating programs will take care of that.
Q: Where can I find an explanation of the internal format
used in the database? The %I, %S, etc. lines?
A: See the
internal format page.
Q: I notice that sometimes you give
more than three lines of terms for a sequence.
What's the story?
A: There is no definite limit on the number of terms that
are given in the database.
The sequence of numbers of meanders (A005316) for example, is exceptionally long. This is because the sequence is interesting, the terms are fairly difficult to compute, and so it seems worthwhile giving as many as possible.
The editing programs will normally truncate the sequence to three lines (roughly 180 to 210 characters including the separating commas, depending on the program), but they can be overruled. If you feel the sequence is important enough to justify this, please add a note to that effect in the Comments.
If fewer than three lines worth are given, feel free to compute some more terms!
Q: What's the preferred
way to enumerate array elements to get an OEIS sequence?
A: In the case of an infinite square array,
a11 a12 a13 ... a21 a22 a23 ... a31 a32 a33 ... a41 a42 a43 ... ...............this normally gets read by antidiagonals and recorded as the sequence
a11 a12 a21 a13 a22 a31 ...
or
a11 a21 a12 a31 a22 a13 ...
(pick whichever seems nicer - or use both). See A003987 for an example.
Q: How should an array or triangle
be included with the submission?
A: Put it in the "Example"
(or %e) lines.
For instance (A079297):
%I A079297 %S A079297 1,2,6,3,9,15,4,12,20,28,5,15,25,35,45,6,18,30,42,54,66,7,21,35,49,63, %T A079297 77,91,8,24,40,56,72,88,104,120,9,27,45,63,81,99,117,135,153,10,30,50, %U A079297 70,90,110,130,150,170,190,11,33,55,77,99,121,143,165,187,209,231,12 %N A079297 Triangle read by rows: the k-th column is an arithmetic progression with difference 2k-1, and the top entry is the hexagonal number k*(2*k-1) (A000384). %C A079297 The n-th row consists of the odd multiples of n from n*1 to n*(2n-1). %D A079297 R. Honsberger, Ingenuity in Math., Random House, 1970, p. 88. %F A079297 a(n,k) = n(2k-1) for 1<=k<=n. n-th row adds to n^3. %e A079297 Triangle begins: %e A079297 1 %e A079297 2 6 %e A079297 3 9 15 %e A079297 4 12 20 28 %e A079297 5 15 25 35 45 %e A079297 6 18 30 42 54 66 %K A079297 nonn,tabl,easy %O A079297 1,2 %A A079297 njas, Mar 04 2003
Q: Some of the entries have figures or other
files associated with them. What's your policy on this?
A: As long as the files are not too big and
serve a useful purpose they are welcomed.
Examples of such files are:
The reason for including these files on the OEIS web site is that it is hoped that this is more permanent than people's individual home pages.
Q: Questions concerning decimal expansion of constants.
The OEIS contains many sequences giving the
decimal expansions of important constants.
For example,
the decimal expansion of Pi, 3.14159265358979...,
gives the sequence
3,1,4,1,5,9,2,6,5,3,5,8,9,7,9,...
(A000796),
with offset 1.
Q: How many new sequences and comments come in each day? How big is the database?
A: The number of new sequences arriving has remained
fairly constant at about 10000 per year (i.e. 30 a day) for several years.
The number of comments keeps increasing, and at present averages between 30
and 60 a day. Web traffic on all my web pages averages
about 600000 page-downloads per month.
The number of sequences in the OEIS is posted on the main Lookup page and is constantly updated. As of January 2006 it is around 120000. The database (just the entries for the sequences, not the illustrations) occupies about 100 megabytes.
Q: What can I do to help?
A:
There are a large number of other journals available on the Web that often contain sequences, but for which you (or your institution) need a subscription. The European Journal of Combinatorics, the Journal of Mathematical Chemistry are just two of dozens that could be listed here. If you have access to a good library, you should be able to search these journals for new sequences or new references for existing sequences.
Q: You asked people to
help edit sequences with keyword "uned". What exactly do you need done?
A:
It is not easy to give a precise answer.
The best thing would be to look at some well-written entries in the OEIS (here are some picked at random off the WebCam: A001316, A055545, A052402, A007308).
Then look at the two web pages that describe the internal format and the standard or beautified format used in the replies from the lookup service.
This is how things should be!
Now look at some entries with keyword "uned". (Search here.)
Usually you will see that many things are wrong! The description is obscure, or the entries are obviously wrong, or the English is hopeless, etc.
To help, make a copy of the sequence in the internal format, edit it (very carefully) and email it to me (njasloane@gmail.com) using subject line "EDITED A012345" (say).
Sequences that need a lot of work have keyword "uned", but there many others that could be improved.
If you don't see anything wrong, send me email saying sequence A012345 (say) looks fine and doesn't need further editing.
Q: I am emailing you a corrected version of a sequence. Should I say "Edited by ...", "Extended by ...", or not sign it at all?
A: If you just added more terms, say something like:
%E A060031 More terms from Larry Reeves (larryr(AT)acm.org), Jan 01 2003 %E A070171 Corrected and extended by Ralf Stephan (ralf(AT)ark.in-berlin.de), Feb 02 2002If you made some nontrivial changes, say something like:
%E A067581 Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Mar 03 2002 %E A069841 Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 04 2002 %F A083741 G.f.: 3*(x-2)*ln(1-x)-5*x+x^2. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 06 2003 %F A006721 a(n) is asymptotic to C^n with C=1.226....... - Benoit Cloitre (abcloitre(AT)wanadoo.fr), Aug 07 2002Reasons for doing this:
Q: How can one obtain a file of all the sequences (stripped of formatting) for running tests?
A: Go to the
Welcome page,
and scroll down. After a while you will
see a link to a stripped-down and gzipped version of the database.
Q: Why are there no mirror sites?
A: This is essentially a one-man operation,
and it is hard enough to maintain one site.
Q: Can one subscribe to the comment mail queue?
A: Not at present.
But unless I am traveling, comments usually get processed within
a few days.
Thanks to all the sequence fans who suggested questions and answers for this page.