If you are interested in working on any of these,
please feel free to do so!
My email address is
Want to Help?
The following is a list of some projects that need doing
in connection with the
All help will of course be acknowledged.
Last major update was in 2005; minor updates Dec 15 2012; Jul 14 2016.
Comments and suggestions for additional entries are welcomed.
Update the references to web pages.
In many cases the entry for a sequence just gives a link
to a web page, without mentioning the author of
the page, or even the title. Many simply say "Source", or "More information".
Typical examples (with spaces
inserted to make the line visible) are:
%H A002457 < a href= " h t t p : / / m a t h s t a t . c a r l e t o n . c a / ~ z h a o / T E A C H I N G / 7 0 . 2 6 5 / r a n d o m - v / r a n d o m - v . h t m l " >Source< / a >
%H A000396 < a href = " h t t p : / / w w w . u t m . e d u / r e s e a r c h / p r i m e s / m e r s e n n e . s h t m l " >Perfect numbers< / a >
What needs to be done:
Follow every link, make a note
of the authors' names and the title of the page,
and construct a better link and send it to me.
The preferred format is:
%H A060638 M. Latapy, < a href = " h t t p : / / w w w . l i a f a . j u s s i e u . f r / ~ l a t a p y / Z o n o / i n d e x . h t m l " >Tilings of zonotopes< / a >
%H A000396 J. S. McCranie, < a href = " h t t p : / / w w w . r e s e a r c h . a t t . c o m / ~ n j a s / s e q u e n c e s / J I S / i n d e x . h t m l # P 0 0 . 1 . 3 " >A study of hyperperfect numbers, J. Int. Seqs. Vol. 3 (2000) #P00.1.3< / a >
with the author's name first, then the title, etc., inside the link.
The reasons for doing this are: to give more information about the web pages, to give more credit to the authors, and to make these web references have the same format as references to books and journals. There's no reason to distinguish between the different types of references.
If a link is broken, try to find out the correct link, if necessary
by contacting the author directly.
Many electronic journals are available on the web these days,
as well as the collected papers of scientists both living and
dead. See for example the web site of the
European Mathematical Information Service (EMIS).
Furthermore a large number of preprints in mathematics and physics
are available from web sites such as the
LANL eprint arXiv.
What needs to be done:
Scan these journals, books and preprints looking
for new sequences or additional references for existing sequences.
Here are some journals that very often contain integer sequences
(but which I do not have time to scan):
Suggestions for other journals to add to this list will
When you see a sequence in any of these sources,
send it in using the
Web page for submitting sequences.
Give your name as the author, and give the source in the reference or links
Even if you don't fully understand the definition,
you can always say something like: "Related to
the enumeration of poly-fusenes".
Believe me, the next person who comes looking for this sequence
will be very grateful for the reference, even if the description
is not very precise.
Help spread the word about the OEIS
People are constantly telling me that they just found
out about the Encyclopedia, and how they wished they had
known about it years ago,
Many university departments, libraries, individuals, etc.,
publish lists of useful web sites. It would be nice
if more of them would mention the On-Line
Encyclopedia of Integer Sequences.
What needs to be done:
For any list of "useful sites" or "helpful links"
that you can find on the Web, send them a message proposing
that they include a link to the OEIS.
Typically the following information is what they need:
- Name: On-Line Encyclopedia of Integer Sequences
- URL: http://www.oeis.org/
- Brief description: Have you ever come across a number
sequence in your work (or play)
- such as 1, 1, 2, 3, 6, 11, 23, 47, ... -
and wanted to find out what was known about it (or even simply
the next term)? This is the place to find out.
- Category: Mathematics (or Science) reference
- Contact person: N. J. A. Sloane (email@example.com)
More information about asymptotics.
What needs to be done:
- Andrew Odlyzko's
Asymptotic enumeration methods, pp. 1063-1229 of R. L. Graham et al., eds., Handbook of Combinatorics, 1995
also the publications of Edward A. Bender, Daniel J. Kleitman
(look them up in MathSciNet), and others;
also any sequences that have references to "combstruct",
(The Combstruct database often gives the asymptotic behaviour
of the sequences that it mentions.)
For any sequences discussed in these references,
see if there is a formula known for the
asymptotic behaviour of a(n) when n is large.
If there is, mention this:
%F A123456 For asymptotic behavior see the article by J. Smith, ...
(and of course add the reference if it is not aleady present).
If the formula is simple enough, then please give it:
%F A123456 For large n, a(n) is asymptotic to 3*2^n - see the article by J. Smith...
Sequences from generalized polyominoes.
There are a number of sequences concerned with counting
polyominoes of various kinds where the description
needs to be made more precise.
Some examples are
What are these sequences?
Many of them are taken from web pages like these:
Livio Zucca's page about pentominoes
What needs to be done? Give a precise definition.
Borrow or create a nice picture (as a gif or jpg file, say)
to illustrate one of the early terms (see
for an example).
Track down similar sequences by following the links in these pages.
More illustrations for important sequences.
- note the link to a figure that
illustrates one of the early terms in the sequence.
I would like to get such illustrations (whenever they
are appropriate) for all the important sequences,
especially those with keyword "core".
Sometimes illustrations can be found on the web.
Often it is possible to get permission to use someone else's
picture. Typically what I do is the following.
Say Jane Smith has a nice web page about caterpillars.
Add a line in the entry for that sequence in the database saying something like:
%H A123456 <a href="http://www.xyz.edu/~smith/">Jane Smith's page about
Make a copy of her gif or jpg file that shows the caterpillars of order 4,
and link to it by a line like:
%H A123456 <a href="a123456.gif">
Jane Smith's picture of caterpillars of order 4.</a>
Of course I ask permission first. I have found that people
are usually very willing to cooperate.
The reason I prefer to make a copy of the gif file and put it
on the sequence web site, rather than just
putting in a link, is that URL's change unexpectedly.
Someone said that the average life of a web site is about 6 months.
The address of the sequence web site has been unchanged for at least five years.
Henry Bottomley (se16(AT)btinternet.com),
Vladeta Jovovic (vladeta(AT)Eunet.yu),
who have contributed illustrations. Much more remains to be done however.
Look for expired links
I have a program that I run every so often that
looks to see if any of the links in the database have expired.
Usually this turns up a large number of dead links.
Maybe someone would be willing to look
at some of the dead links,
and do the detective work necessary
to find out if they have moved or if they have
been deleted from the Web. If the latter,
people are sometimes willing to allow the OEIS
to install a copy on the OEIS web site.
(Perhaps the person has moved and no longer has a home
page of their own.)
The list of bad links as of Dec 13 2000 can
What needs to be done. For each link on that list:
- Check that it is indeed broken (I may have just been unlucky)
- find the one or more sequences that mention it;
- then track down what happened to it
- send in the corrected link (or let me know
if it is hopelessly lost) by filling out the
Comment on an existing sequence form.
- The most serious thing that can happen is that the only
known description of a sequence is on the missing web page.
In this case it is important to try to recover
the definition of the sequence!
Thanks mostly to the work of James Sellers
the list of bad links is now empty.
Many thanks, James!
More Maple or Mathematica programs for important sequences.
Many entries have programs in Maple, Mathematica, Pari,
Magma, etc. which generate the sequence.
I would like to get such programs (whenever they
are appropriate) for all the important sequences,
especially those with keyword "core" or "nice".
More Engel expansions
Take a look at
. It would be nice
to have Engel expansions for more of the standard
constants (sqrt(2), the golden mean, etc.).
I did not look to see what expansions are there now
file for this) but there are not many I believe.
Jan. 17, 2001: Many thanks to Mitch Harris (maharri(AT)cs.uiuc.edu)
who has followed up on this suggestion. See the expanded list of
sequences related to Engel expansions in the Index file.
Sequences from binomial coefficient identities.
I would like to get lots more sequences like this.
What needs to be done:
Pick one of the standard
references that gives lists of binomial coefficient
For a sum that depends on a single parameter n
(such as that in
) convert it into
a sequence (and give both sides of the identity).
For a sum with two parameters m and n, convert it into
a triangular or square array and hence into
a linear sequence by reading the
array by rows or by antidiagonals.
See the file
Welcome to the On-Line Encyclopedia of Integer Sequences
for more information about
converting arrays into sequences.
If the values of the left- and right-hand sides are integers,
then that's the sequence.
If they are fractions, then as usual enter the numerators
and denominators as separate sequences, with keyword "frac"
and linked together by cross-references.
H. W. Gould's "Combinatorial Identities" is the obvious place
to start. I have added a handful of sequences from it,
but have not gone through it in any systematic way.
Please give references for the binomial coefficient identities
that you are using.
Dec. 29, 2000: Many thanks to Yong Kong
(ykong(AT)curagen.com), who has sent in many new sequences or comments
on existing sequences based on binomial coefficient identities
from the book A. P. Prudnikov, Yu. A. Brychkov, and O.I. Marichev,
"Integrals and Series", Volume 1: "Elementary Functions",
Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
Sequences from inequalities.
I would like to get lots more like this sequence
(see also previous item).
D. S. Mitrinovic et al.'s Handbook of Number Theory, Kluwer,
is a good source for inequalities in number theory.
I have extracted a handful of
sequences from it but have not gone through it in any systematic way.
Please give references for any inequalities that you use.
Sequences from discrete math or other books.
Pick your favorite book on combinatorics, discrete math.,
number theory or other branch of science,
and make a concordance file listing all the sequences in it,
whether explicit or implicit - from the solutions
to problems, for instance.
See the files giving the sequences in
Comtet's Advanced Combinatorics,
Harary and Palmer's Graphical Enumeration
Stanley's Enumerative Combinatorics
for examples of such concordances.
What needs to be done: ideally, work through the book,
do all the problems, and keep track
of all the sequences that occur.
(Remember that sequences of rationals numbers are handled
by giving separate sequences for
the numerators and denominators.)
For each sequence found, make sure it is in the database,
send it in as a new sequence if it isn't,
with a reference to the book, of course.
If the sequence is already in the database,
check that there is a reference to the book
from that sequence, and if
there isn't, send that in as a
Comment on an existing sequence.
Some books that could be treated in this way are the following.
(These are books from which I have already extracted many sequences.)
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings)
(Several electronic versions of this book now exist -
it would be nice to have links to them.)
- F. Bergeron et al., Combinatorial Species and Tree-Like Structures, Camb. 1998.
- E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982
- T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949.
- J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 3rd edition.
- R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd edition.
- R. K. Guy, Unsolved Problems in Number Theory, second edition.
- F. Harary, Graph Theory, Addison-Wesley, 1969.
- G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers [since there are many editions of this book, if possible references should avoid mentioning page numbers.
D. E. Knuth,
The Art of Computer Programming, 3+ Vols. Of course
it would be too big a job to go through every page of this work looking for sequences.
But even a partial concordance would be useful.
Many more sequences are waiting to be extracted from these books.
Since there are many different editions of each volume on people's
shelves, if possible references should be written
so as to refer to chapters, sections, equation numbers and paragraphs,
rather than to page numbers.
- J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958
- J. Riordan, Combinatorial Identities, Wiley, 1968
Sequences from the physics literature.
More than 20 years ago I scoured all the main physics journals
(J. Math. Phys, J. Phys. A, etc etc) and found
many sequences. This needs to be done again!
Of course for recent papers one can also check the LANL web site.
Sequences from the chemistry literature.
More than 20 years ago I scoured the chemical journals
like J. Math. Chem. and found
many sequences. This needs to be done again!
The relatively recent journal MATCH especially needs to be checked.
What needs to be done: Look at all the issues of the
appropriate journals, MATCH, Combinatorial Chemistry,
J. Math. Chem., etc.
Also look for papers in the indexes to the chemical
literature for papers with "number of" or
"enumeration" in the title or abstract.
Note in particular that Combinatorial Chemistry
can be accessed freely if one is a subscriber to
(and that is also free).
On Dec 13 2000 Elemer Labos (labos(AT)ana1.sote.hu)
made the following suggestions:
"Concerning chemical items I know of a group of
German and Croatian authors
who published a large amount of Computer Generated Organic
Molecules ... data (isomer enumeration in the style of Cayley or Polya).
"This was in the Cayley -- Polya tradition.
In 1985 they published a lot of isomer enumeration data.
Not only trees, rooted trees, physical trees but also
condensed hex-es of the most diverse types.
I am sure you could find new things in their work.
"A very interesting paper is
Knop JV, Mueller WR, Szymanski K, Trinajstic N,
Computer generation of certain classes of molecules.
SKTH/Kemija u industrij, Zagreb, 1985,
"I found Nenad Trinajstic on the Internet as follows:
CROATICA CHEMICA ACTA Editor in Chief - N e n a d T r i n a j s t i c
Editor-in-Chief of Croatica Chemica Acta since 1994 N. Trinajstic
is a senior scientist at Rudjer Boskovic Institute, full prof. at
Faculty of Science , University of Za
CROATICA CHEMICA ACTA 72 (1999) No. 1 - CROATICA CHEMICA ACTA 72
1 Croat. Chem. Acta Vol. 72 No. 1 I-II, 1-133, A1-A63, B1-B5,
C1-C6 (1999) Zagreb, March, 1999 CONTENTS Editorial . . . Nenad
Trinajstic I-II Physical and
"I think they would be
happy to send you their 1985 publication,
including a lot of graphical enumeration data.
"Also I found a coauthor of Trninajstic perhaps in USA ??
DR. EMIL POP'S PUBLICATIONS
1. N. Bodor, E. Pop and N. Trinajstic, "Valence Shell and p Electron
SCFMO Calculation for the Isomerism of the
21-Benzylidene-20-oxo-pregnane Derivatives." Rev. Roumaine Chim., 16,
2. L. Klasinc, E. Pop, N. Trinajstic and J.V. Knop, "Theoretical
Studies of Positional Isomers Obtained by Annelation of Benzene and
5-Membered Ring Heterocyclics Containing N, O or S." Tetrahedron, 28,
3. L. Klasinc, N. Trinajstic and E. Pop, "Theoretical Study of
Isomeric Thienylfurans." Rev. Roumaine Chim., 18, 89-98 (1973).
4. E. Pop, N. Trinajstic and L. Klasinc, "Theoretical Studies of Some
Furocoumarine Isomers." Rev. Roumaine Chim., 18, 1249-1257 (1973).
115. E.Pop, "Water-Soluble Combinations of Dexanabinol: Prodrugs and
Analogs", Die Pharmazie, 55, 000-0000 (2000).
Alchem Laboratories Corp Tel: (904)418-1650 FAX: (904)418-1584
13305 Rachael Boulevard, Alachua, Florida 32615 U.S.A., info(AT)alchem.com
"I think this Dr Pop can give information on how to contcat Trinajstic
and get (I am sure in) intersting raw material for EIS.
"Their polyhex isomers seem to me 2D dominoes or animals .
"The German coauthor J.V. Knop was 15
years ago in University of Duesseldorf , Computer Center.
I could not find him through the Internet.
[End of quotation]
Mysterious or badly-described sequences
If a sequence has the
with keyword "obsc", for "obscure", it means I was unable to
understand the description.
The keyword "uned", for "unedited", means that I did not have
the time to edit the entry in the usual way, which is to:
- check that the description is mathematically sensible
- check that all the lines are in correct English
- check that all the lines have the appropriate labels - any Maple
lines begin %p, any Mathematica lines begin %t, etc.
All such sequences need to be cleaned up.
What needs to be done: for any sequence that has keyword "uned" or
"obsc", try to understand the description of the sequence,
rewrite the description more clearly and in correct English,
and polish up all the other lines in that entry.
Quite often I get correspondence about sequences that
calls for a fair amount of thought to understand.
For example, complicated messages discussing whether
two sequences are really the same. It is great fun
working through such material, but I'm willing to share the fun,
if there are volunteers.
This sequence (and any others from the same source)
needs a better reference (and a better description if possible):
%S A037294 2,7,18,42,90,186
%N A037294 Number of (s,2) gates.
%D A037294 E Detjens and G Gannot, Technology mapping in MIS,
pp. 116-119 of some 1987 IEEE conference proceedings [ # CH2469-5/87 ].
%K A037294 nonn
%O A037294 1,1
%A A037294 N. J. A. Sloane (njas(AT)research.att.com).
Improve the index.
to the database needs a lot of work.
What needs to be done: pick a topic that interests you and make an
index to all the sequences that deal with that topic.
For an example, look at the entries under
in the index.
Christian G. Bower (bowerc(AT)usa.net),
Mitch Harris (maharri(AT)cs.uiuc.edu),
Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de),
who have contributed sections to the index.
Much more remains to be done however.
In a number of cases the references for a sequence
just give a volume and page number but not the author or title, etc.
(for a typical example see
A033874 - there are lots more like this).
In other cases
there are references saying things like "[Ii]n [Pp]reparation",
"[Ss]ubmitted", "[Tt]o [Aa]ppear", "[Ii]n [Pp]ress", etc.
What needs to be done:
Update all the incomplete or out-of-date references.
Ideally every reference should give full details. Here is the preferred format:
A. B. Smith, Title of paper, Name of Journal, Vol. 66 (2000), 34-45.
J. H. Smith, Problem 56, Name of Journal, Vol. 66 (2000), 34-45.
M. K. Smith, Title of Book, Publisher, Place, Edition, Year; see especially Eq. (100) on page 564.
Update the French version
The French version is quite out of date.
Originally it was a direct translation of the English version.
But since then I have completely changed the English version
What needs to be done is to translate the English
versions of the current pages into French, and completely redo
the French version, making it parallel to the English
There is a
of sequences that need extending.
- Links to MathSciNet
I wish it were possible to have a link to MathSciNet
whenever the entry for a sequence cites a paper or book that has
been reviewed by Math. Reviews.
However, since MathSciNet is a commercial service - and
so not accessible to the general public - I don't see
hope of doing this at present.
Hopefully one day it will be freely available.
- General remarks
Note that if you are contemplating doing serious work
on the database, it is possible to download
any section of it by going to the file
Welcome to the On-Line Encyclopedia
If you have occasion to write to someone while working
on one of these projects, feel free to say something like
"I'm helping Neil Sloane with the On-Line
Database of Integer Sequences and the following question
has arisen in connection with your sequence A****** ..."