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- Contents of this page:
New Users
Description of the Database
Sources
Editorial Board
Arrangement of Sequences in Database
The Full Database
Gzipped Version
Gzipped Names
Contributing New Sequence or Comment; Helping
Sequences in Classic Books
Papers Citing the Encyclopedia of Integer Sequences
Referencing the OEIS
URLs
Referencing a Particular Sequence
URL for Searching the Database
Policy on Email Addresses
Copyright Notice
Acknowledgments
Links
Awards, etc.
-
New Users:
- Let's begin at once with an example of a sequence of great importance:
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A060843 |
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Busy Beaver problem: maximal number of steps that an n-state Turing machine can make on an initially blank tape before eventually halting. |
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+30
5
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OFFSET
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1,2
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COMMENT
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"In 1965 [Tibor] Rado, together with Shen Lin, proved that BB(3) is 21. ... Next, in 1983, Allan Brady proved that BB(4) is 107. ... Then, in 1989, Heiner Marxen and Juergen Buntrock discovered that BB(5) is at least 47,176,870. ... As for BB(6), Marxen and Buntrock set another record in 1997 by proving that it is at least 8,690,333,381,690,951." Aaronson.
The function Sigma(n) (A028444) denotes the maximal number of tape marks which a Turing Machine with n internal states and a two-way infinite tape can write on an initially empty tape and then halt. The function S(n) (the present sequence) denotes the maximal number of steps (shifts) which such a machine can make (it needs not produce many tape marks).
Given that 5-state machines can compute Collatz-like congruential functions (see references), it may be very hard to find the next term.
The sequence grows faster than any computable function of n, and so is non-computable.
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REFERENCES
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Brady, A. H., The busy beaver game and the meaning of life, in Herkin, R. (Ed) The Universal Turing Machine, pp. 259-277, Oxford Univ Press 1988.
Brady, A. H. The determination of Rado's noncomputable function Sigma(k) for four-state Turing machines, Math. Comp. 40 #62 (1983) 647-665.
Machlin, R. (nee Kopp), and Stout, Q, The Complex Behavior of Simple Machines, Physica D 42 (1990) 85-98
Michel, Pascal, Busy beaver competition and Collatz-like problems, Arch. Math. Logic (1993) 32:351-367.
R. M. Robinson, Minsky's small universal Turing machine, Int'l Jnl. Math, 2 #5 (1991) 551-562.
Yu. V. Rogozhin, Seven universal Turing machines (Russian), abstract, Fifth All-Union Conference on Math. Logic, Akad. Nauk. SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1979, p. 127.
Yu. V. Rogozhin, Seven universal Turing machines (Russian), Systems and Theoretical Programming, Mat. Issled. no. 69, Akademiya Nauk Moldavskoi SSSR, Kishinev, 1982, pp. 76-90.
Claude E. Shannon, A universal Turing machine with two internal states, Automata Studies, Ann. of Math. Stud. 34 (1956) 157-165.
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LINKS
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Scott Aaronson, Who Can Name the Bigger Number?
Bill Dubuque, Re: Halting is weak
A. Gravell and U. Ultes-Nitsche, BB(n) Grows Faster Than Any Computable Function
H. Marxen, Busy Beaver Problem
M. Somos, Busy Beaver Turing Machine
M. Somos, Busy Beaver
Q. F. Stout, The Complex Behavior of Simple Machines
E. W. Weisstein, Link to a section of The World of Mathematics.
E. W. Weisstein, Busy Beaver
Index entries for sequences related to Busy Beaver problem
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CROSSREFS
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Cf. A028444.
Sequence in context: A012662 A012418 A083558 this_sequence A026650 A009253 A012840
Adjacent sequences: A060840 A060841 A060842 this_sequence A060844 A060845 A060846
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KEYWORD
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hard,nice,nonn,bref
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AUTHOR
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Jud McCranie (j.mccranie(AT)adelphia.net) and njas, May 02 2001
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EXTENSIONS
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The next two terms are at least 47176870 and 3*10^1730.
Additional references from Bill Dubuque (wgd(AT)martigny.ai.mit.edu)
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- Most people use this web site to
get information about a particular number sequence. If you are a new visitor,
then you might ask the database if it can recognize your favorite sequence,
if you have one. To do this, go to the
main look-up page.
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-
If your sequence isn't in the database, and if it is interesting,
please submit it using the web page for
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If you have stumped the database, you can try
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You can browse the database, using the
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It is interesting to scan the
Index to
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In a way this database can be regarded as an index to all
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-
Also worth visiting are the pages dealing with
Puzzle sequences,
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Hot sequences.
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You can run the
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The Search pages have little buttons called "list",
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and sometimes "table".
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"List" produces a numbered list of the terms,
plus a bracketed list suitable for importing into other programs.
-
"Graph" produces two plots of the sequence. The first is a pin plot
of the first 200 terms (less if fewer terms are available), the
second is a linear or log scatterplot of all available terms, using
terms from the b-file if there is one.
Some noteworthy plots are
the Fibonacci numbers, A000045,
the partition numbers, A000041,
the Euler phi-function, A000010, etc.
The plotting program was written by Deborah Swayne using the R language.
-
"Listen" produces a midi file so that you can listen to the sequence.
The first time you use it you will probably have to
tell your browser to allow popups from the OEIS web site.
(This works best with Firefox.)
Try listening to Recaman's sequence
A005132, turn the volume up to 127
and set the instrument to #103 !
-
"Table": If the sequence is formed by reading a triangle across rows
(or by reading a table by antidiagonals),
this button produces three different two-dimensional views of the sequence.
For an example, see Pascal's triangle A007318.
-
Finally, you might like to see a
list of papers
that have acknowledged help from the database
and some
comments from readers.
- Description of the Database (or, What is the Next Term?)
What comes next after
1, 2, 4, 9, 20, 48, 115, 286, 719, ... ? (for example). This is the place to find out!
The main table is a collection of number sequences arranged in lexicographic order.
The entry for each sequence gives:
- the beginning of the sequence
- its name or description
- any references or links
- any formulae
- cross-references to other sequences
- the name of the person who submitted it, etc.
For further information about the format of replies received from the database,
click here.
A second file describes the internal format
in which the sequences are stored in the database.
See also the hints file
for further useful information.
- Sources:
Since the mid-1960's
Neil Sloane has been collecting integer sequences from every possible
source. His goal is to have all interesting number sequences in the table.
At the present time the table contains over 100000 sequences.
5487 of the best of these sequences were published in 1995 in
The Encyclopedia of Integer Sequences,
by Neil Sloane and Simon Plouffe.
The book is still useful,
since it contains many of the most important sequences.
The database (which would now fill 750 volumes the size of the 1995 book)
is too huge to use except as a reference.
- Editorial Board:
Beginning in 2002, a group of associate editors has been
helping to process new sequences and updates to the database.
At present the associate editors are:
- Max Alekseyev (maxal(AT)cs.ucsd.edu)
- David Applegate (david(AT)research.att.com)
- Christian G. Bower (bowerc(AT)usa.net)
- Ray Chandler (RayChandler(AT)alumni.tcu.edu)
- Russ Cox (rsc(AT)swtch.com)
- Emeric Deutsch (deutsch(AT)duke.poly.edu)
- Richard K. Guy (rkg(AT)cpsc.ucalgary.ca)
- Maximilian Hasler (maximilian.hasler(AT)gmail.com)
- Dean Hickerson (dean(AT)math.ucdavis.edu)
- Antti Karttunen
- John W. Layman (layman(AT)math.vt.edu)
- Marc LeBrun (mlb(AT)well.com)
- R. J. Mathar (mathar(AT)strw.leidenuniv.nl)
- Jud McCranie (j.mccranie(AT)adelphia.net)
- Tony Noe (noe(AT)sspectra.com)
- Simon Plouffe (plouffe(AT)math.uqam.ca)
- Don Reble (djr(AT)nk.ca)
- Neil J. A. Sloane (njas@research.att.com), editor-in-chief
- Michael Somos (somos(AT)grail.cba.csuohio.edu)
- Stefan Steinerberger (stefan.steinerberger(AT)gmail.com)
- David Wasserman (dwasserm(AT)earthlink.net)
- David W. Wilson (davidwwilson(AT)comcast.net)
- Robert G. Wilson v (rgwv(AT)rgwv.com)
Many other volunteers help by sending corrections, comments,
links or even completely editing an entry.
- Arrangement of Sequences in Database. Most of the sequences are arranged
in the database in lexicographic order of absolute values,
indexed by the position of the first term
that is greater than 1 in absolute value.
Sequences that contain only 0's, 1's and -1's are in strict
lexicographic order by absolute value at the beginning of the table.
Thus there is an essentially unique place to look in order to
see if a sequence is already in the table. (If it
isn't, submit it and it will added if it is sufficiently interesting
- see Sending in a new sequence.)
Sequences received in the last few days and
not yet placed in the lexicographic ordering will be found
at the end of the table.
- Format used in replies from the database.
Internal format used in the database.
- Short index to
the most important sequences.
The main look-up page
will also allow you to search for a word (or do much more
complicated searches) in the database.
- The Full Database
(arranged lexicographically, as explained above)
is contained in the following files.
Note that these are all very large files!
Recent arrivals are at the end of the last file.
The list of files
and the first sequence in each file are as follows:
Part 0 :
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
...
Part 1 :
1,1,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,0,
...
Part 2 :
1,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,
...
Part 3 :
1,0,0,1,0,0,2,0,1,2,0,1,4,0,2,4,1,2,6,1,4,6,2,4,10,2,6,
...
Part 4 :
0,0,0,1,2,0,4,3,2,1,0,1,0,3,0,1,2,7,4,3,0,1,2,0,3,2,0,3,8,3,0,1,2,4,0,
...
Part 5 :
0,1,0,2,1,0,3,2,1,0,4,3,2,1,0,5,4,3,2,1,0,6,5,4,3,2,1,0,7,6,5,4,3,2,1,0,
...
Part 6 :
1,1,1,1,2,1,1,1,2,7,1,1,1,3,13,3,938,41
...
Part 7 :
1,1,1,1,2,1,1,3,4,1,1,4,11,7,1,1,5,26,32,11,1,1,6,57,122,76,16,1,1,7,
...
Part 8 :
1,2,1,2,2,1,0,0,0,1,0,0,0,2,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,
...
Part 9 :
1,2,1,3,0,4,4,3,0,4,5,0,0,0,16,6,3,8,0,0,4,7,0,0,0,0,0,36,8,7,0,12,0,0,
...
Part 10 :
1,2,1,4,1,1,7,2,1,1,12,3,1,1,1,19,5,2,1,1,1,30,7,3,1,1,1,1,45,11,4,2,1,
...
Part 11 :
1,2,1,7,2,2,24,8,1,103,1,12,43,94,21,1,11,12,4,23,27,4,89,20,42,1,43,6,
...
Part 12 :
1,1,1,2,2,1,1,6,6,3,5,2,1,1,24,24,12,20,14,10,7,5,2,1,1,120,120,60,100,
...
Part 13 :
1,1,1,1,1,2,2,2,2,2,2,3,3,3,6,6,6,11,11,22,22,22,22,41,41,41,41,114,
...
Part 14 :
1,2,2,2,3,6,14,36,99,286,858,2652,8398,27132,89148,297160,1002915,
...
Part 15 :
1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,11,12,
...
Part 16 :
1,2,2,3,9,3,4,24,24,4,5,50,100,50,5,6,90,300,300,90,6,7,147,735,1225,
...
Part 17 :
0,1,2,2,5,2,6,4,6,3,12,2,10,6,8,4,13,2,18,6,10,4,16,4,12,9,12,4,26,2,
...
Part 18 :
1,0,1,2,3,0,12,40,100,0,1225,6860,28812,0,1037232,9779616
...
Part 19 :
1,2,3,2,5,8,8,4,7,2,7,4,7,2,9,6,8,8,8,8,5,8,6,2,10,4,1,4,5,6,5,10,4,4,
...
Part 20 :
1,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,1,2,3,4,3,2,1,0,0,
...
Part 21 :
0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,18,20,22,24,26,28,30,32,33,36,
...
Part 22 :
1,2,3,4,5,6,7,8,10,11,12,13,14,16,17,18,20,21,22,24,26,28,30,31,32,36,
...
Part 23 :
1,2,3,4,5,7,8,9,16,6,11,13,17,19,23,25,27,32,49,53,64,81,128,256,512,
...
Part 24 :
1,2,3,4,6,8,9,10,12,14,5,7,15,16,18,20,21,22,24,25,26,11,13,27,28,30,
...
Part 25 :
1,2,3,4,10,12,28,32,144,480,1056,1152,4992,5376,11520,36864,208896,
...
Part 26 :
0,2,3,5,6,9,11,18,21,23,27,29,30,32,36,42,44,48,56,59,60,62,63,65,66,
...
Part 27 :
2,3,5,7,11,101,131,151,181,191,313,353,373,383,727,757,787,797,919,929,
...
Part 28 :
1,2,3,5,10,15,25,50,75,125,250,375,625,1250,1875
...
Part 29 :
1,2,3,6,9,16,27,47,81,141,243,421,729,1263,2187,3788,6561,
...
Part 30 :
1,1,2,3,7,14,25,46,97,189,344,674,1383,2683,4950,9955,20175,
...
Part 31 :
0,1,2,3,11,22,33,101,111,121,131,202,212,222,232,303,313,323,333,1001,
...
Part 32 :
1,1,2,4,1,8,4,3,16,12,13,8,3,32,32,42,38,33,15,10,1,64,80,120,133,145,
...
Part 33 :
0,2,4,5,7,9,10,12,14,16,17,19,21,22,24,26,28,29,31,33,34,36,38,40,
...
Part 34 :
0,0,2,4,6,9,1,3,5,8,0,2,4,6,9,1,3,5,8,0,2,4,6,9,1,3,5,8,0,2,4,6,9,
...
Part 35 :
1,1,2,4,7,13,23,42,77,132,241,438,742,1348,2435,4102,7418,13341,
...
Part 36 :
1,2,4,8,16,96,64,864,72,1536,252,69120,4096,86400,4608,256,768,24576,
...
Part 37 :
1,2,4,11,22,44,121,242,514,1331,2662,5654,14641,32612,65524,
...
Part 38 :
1,2,4,792,51648,53824,95328
...
Part 39 :
0,1,1,0,2,5,8,8,0,21,55,89,89,0,233,610,987,987,0,2584,6765,10946,10946,0,
...
Part 40 :
1,2,5,12,26,59,133,301,680,1538,3475,7853,17747,40106,90637,204833,
...
Part 41 :
2,5,19,71,97,103,113,157,163,191,193,211,317,359,373,439,443,467,479,
...
Part 42 :
0,2,6,8,15,18,28,32,36,40,55,60,78,84,90,96,119,126,152,160,168,176,
...
Part 43 :
1,2,6,19,63,216,760,2723,9880,36168,133237,492993,1829670,6804267,
...
Part 44 :
1,2,6,38,16208,47577,78073
...
Part 45 :
2,7,23,71
...
Part 46 :
2,8,18,46,68,120,154,230,368,426,612,760,842,1014,1296,1614,1732,2098,
...
Part 47 :
1,2,9,13,21,80,112,129,147,225,308,349,1063,1282,1300,1635,1880,2686,
...
Part 48 :
1,2,10,100,1700,44200,1635400,81770000,5315050000,435834100000,
...
Part 49 :
0,2,12,118,1880,63274,4010212,332200222,32100167600,3749943333714,
...
Part 50 :
2,16,208,3968,109568,4793344,410662912,82657083392,38274970222592,
...
Part 51 :
1,2,24,160,1232,9120,68224,508928,3799296,28357120,211662848,
...
Part 52 :
2,118,29998,611334362,1141045350238,231711278648170198,
...
Part 53 :
1,0,0,3,0,23,90,721,3864,47879,326214,4585141,40991676,647541971,
...
Part 54 :
3,1,3,3,6,3,6,1,9,0,12,3,6,6,12,3,6,3,12,6,12,6,12,3,15,0,9,6,18,6,18,
...
Part 55 :
1,3,2,1,2,3,1,3,2,3,1,2,1,3,2,1,2,3,1,2,1,3,2,3,1,3,2,1,2,3,1,3,2,
...
Part 56 :
0,1,3,2,7,8,5,4,6,17,20,12,10,15,16,19,11,9,14,22,21,18,13,45,54,31,26,
...
Part 57 :
3,3,5,5,7,7,7,7,11,11,13,13,13,13,17,17,19,19,19,19,19,19,19,19,19,19,
...
Part 58 :
1,3,4,5,6,7,8,9,11,12,13,14,15,16,19,20,21,22,23,24,27,28,29,
...
Part 59 :
0,3,4,7,11,15,18,19,20,23,24,27,35,38,39,42,47,51,55,58,59,66,71,75,
...
Part 60 :
3,5,6,4,4,0,7,2,6,6,0,9,5,4,3,2,5,9,7,7,7,3,5,5,7,5,8,6,5,2,8,9,8,2,4,
...
Part 61 :
3,5,8,9,11,15,21,39,50,63,83,95,99,173,350,854,1308,1769,2903,5250,
...
Part 62 :
1,0,3,5,16,43,113,316,839,2301,6204,16855,45665,123800,335659,909845,
...
Part 63 :
1,3,6,10,15,21,28,36,45,55,66,78,91,171,276,378,1770,2775,5778,8778
...
Part 64 :
1,3,7,3,9,3,4,3,9,2,2,5,0,1,0,6,7,8,9,0,4,0,5,4,2,3,2,3,1,7,8,4,6,
...
Part 65 :
1,3,7,16,29,44,65,89,120,155,192,236,282,332,390,453,520,589,666,746,
...
Part 66 :
1,0,3,8,16,4,60,65,50,5,384,168,462,108,6,2380,763,3836,1624,196,7,
...
Part 67 :
1,3,9,22,55,133,323,780,1885,4551,10989,26530,64051,154633,373319,
...
Part 68 :
1,3,10,39,187,1128,8455,76359,806032,9715773,131479675,1972203654,
...
Part 69 :
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,
...
Part 70 :
1,3,15,63,75,105,165,195,231,255,285,315,495,525,585,735,825,945,975,
...
Part 71 :
1,1,3,20,280,8064,473088,56229888,13495173120,6525665935360,
...
Part 72 :
3,36,32768,20632736881,2096638274713626813358181376
...
Part 73 :
1,4,0,4,9,0,9,1,3,2,7,3,5,7,9,5,5,3,5,5,2,5,4,4,8,1,5,0,6,1,4,6,5,4,3,
...
Part 74 :
1,4,2,4,6,2,2,2,2,6,16,2,6,4,6,6,20,2,4,6,2,2,4,2,2,6,4,12,2,6,2,12,6,
...
Part 75 :
0,1,4,4,12,21,36,81,215,498,1230,3024,7518,18716,47494,119818,
...
Part 76 :
0,0,4,6,4,22,50,118,186,704,1420,3678,8470,23668,54792,146181,
...
Part 77 :
1,0,4,6,107,490,7780,66045,1113185,14056740,264337750,4409272175,
...
Part 78 :
1,4,8,15,21,33,41,56,69,87,99,127,141,165,189,220,238,277,297,339,371,
...
Part 79 :
1,4,9,19,35,52,72,100,131,163,201,244,290,340,393,451,515,580,648,724,
...
Part 80 :
1,4,11,24,39,68,92,118,156,213,268,307,347,432,522,586,651,723,840,969,
...
Part 81 :
1,4,14,39,96,213,437,837,1520,2632,4380,7040,10979,16668,24716,35879,
...
Part 82 :
1,1,4,19,57,178,543,1591,4598,13117,36999,103514,287653,794847,2186054
...
Part 83 :
1,1,4,30,335,4984,92652,2065146,53636520,1589752230,52926799310,
...
Part 84 :
4,141,323,323,667766,1175711,42599524,66411466,158191851,15980308951,
...
Part 85 :
1,5,3,7,2,7,27,8,1,1,9,11,27,27,6,13,72,5,81,5,27,9,99,11,4,27,10,30,
...
Part 86 :
0,5,7,9,10,12,14,15,17,19,20,21,22,23,24,25,27,29,30,31,32,33,34,35,37,
...
Part 87 :
5,9,23,37,39,47,57,97,119,187,257,271,273,281,309,367,449,529,687,759,
...
Part 88 :
1,5,13,29,61,573,2621,6717,23101,88637,350781,875069,9263677,26040893,
...
Part 89 :
5,20,55,125,251,461,791,1286,2001,3002,4367,6187,8567,11627,15503,
...
Part 90 :
1,5,34,201,1179,6909,40488,237256,1390293,8146945,47740079,279750872,
...
Part 91 :
1,5,265,2367,237493,2576561,338350897,616410400171,7811559753873
...
Part 92 :
1,6,6,21,6,36,6,56,21,36,6,126,6,36,36,126,6,126,6,126,36,36,6,336,21,
...
Part 93 :
1,6,12,8,0,12,48,48,15,60,12,96,0,120,240,64,96,234,156,0,0,444,240,96,
...
Part 94 :
1,1,1,6,21,100,645,4256,34153,302352,2966505,32245664,380114493,
...
Part 95 :
1,6,41,309,24722,2060169,176585917,15451267740,1373446021336,
...
Part 96 :
1,6,1221,231880,13443885,340203456,4910472385,47565216504,342540938025,
...
Part 97 :
1,7,9,10,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,31,37,39,41,
...
Part 98 :
7,19,31,43,103,139,151,199,271,283,463,523,571,619,643,811,823,859,
...
Part 99 :
0,1,7,54,413,3161,24192,185149,1417003,10844766,82998377,635212469,
...
Part 100 :
0,0,8,2,8,3,8,3,2,8,5,6,1,3,3,5,9,2,5,3,5,1,2,4,1,3,8,7,2,9,4,4,8,7,2,
...
Part 101 :
8,16,40,56,96,120,176,280,320,456,560,616,736,936,1160,1240,1496,1680,1776,
...
Part 102 :
8,60,850,16380,358258,8253300,194402650,4624699020,110523825058,
...
Part 103 :
1,1,9,3,7,2,1,6,1,4,3,8,3,9,0,0,1,3,7,0,7,3,3,0,2,5,9,3,0,2,1,7,4,
...
Part 104 :
9,25,27,45,81,91,225,249,325,405,481,511,561,645,747,793,891,925,949,
...
Part 105 :
9,163,2943,53137,959409,17322499,312764391,5647081537,
...
Part 106 :
1,10,19,28,55,64,100,127,145,154,163,190,253,280,289,325,379,388,415,
...
Part 107 :
0,10,120,12403,12403,231540,1053426,10623745,120784653,1062489753
...
Part 108 :
11,18,20,25,27,28,32,37,56,81,88,113,115,120,123,125,127,161,162,196,
...
Part 109 :
1,11,111,1111,11111,111111,1111111,11111111,111111111,
...
Part 110 :
1,12,30,46,83,1099,1571,17902874277
...
Part 111 :
1,12,8848,45051072,840261660928,41695924253002752,
...
Part 112 :
1,13,247,5811,159939,4993911,173345913,6593244957,271829934441,
...
Part 113 :
15,21,33,35,55,77,303,393,453,505,543,573,655,707,755,905,917,939,955,
...
Part 114 :
16,32,48,64,80,81,96,112,113,128,129,144,145,160,161,162,176,177,192,
...
Part 115 :
1,17,66,130,194,258,322,386,450,514,578,642,706,770,834,898,962,1026,
...
Part 116 :
19,21,2703,15929,4124583,27067051,179992913,179993011,179993159,
...
Part 117 :
1,20,433
...
Part 118 :
22,250,3750,41250,414150,4166250,42281250,438281250,4400343750,
...
Part 119 :
0,0,0,0,24,60,240,1260,8064,60480,518400,4989600,53222400,622702080,
...
Part 120 :
26,3,12,1,4,1,12,3,52,3,12,1,4,1,12,3,52,3,12,1,4,1,12,3,52,3,
...
Part 121 :
1,28,232,1528,11792,92328,740584,6020560,49474064,409888240,3418201292,
...
Part 122 :
1,30,756,18360,441936,10614240,254788416,6115201920,146766525696,
...
Part 123 :
1,1,33,8506,9483041,33056715626,293327384637282
...
Part 124 :
40,89,138,187,236,250,257,264,271,278,280,281,282,283,284,286,
...
Part 125 :
1,48,1152,18448,221952,2141808,17282432,120037968,733189632,
...
Part 126 :
60,150,600,1500,3390,4320,6000,9240,15000
...
Part 127 :
72,144,192,216,588,600,648,792,936,992,1056,1224,1302,1320,1560,1736,
...
Part 128 :
0,0,0,90,630,2520,7560,18900,41580,83160,154440,270270,450450,
...
Part 129 :
1,112,1136,3136,9328,14112,31808,38528,74864,84784,143136,149184,261184,
...
Part 130 :
151,199,268,367,393,412,477,511,524,537,559,606,622,790,801,863,972,996,
...
Part 131 :
243,371293,6436343,39135393,147008443,418195493,992436543,
...
Part 132 :
407,493,893,1189,1343,1403,1643,1681,1961,3151,3223,4063,4579,7087,
...
Part 133 :
730,4649,9458,15457,15706,22621,31853,32569,41434,53189,54113,56458,
...
Part 134 :
1463,1562,2156,14630,14763,15404,15602,15620,15762,17562,20156,21560,
...
Part 135 :
4096,104976,614656,2085136,5308416,11316496,21381376,37015056,
...
Part 136 :
15783,15927,16879,18266,19466,22292,26186,33806,37668,38333,38432,
...
Part 137 :
262144,524288,2097152,134217728,4294967296,35184372088832,
...
Part 138 :
105639091,105639119,105639143,105639151,105639211,105639223,105639239,
...
Part 139 :
313,337,433,601,673,937,1153,1249,1297,1609,1777,1873,1993,2089,2161,
...
-
The Recent Additions file.
(Note that you can also browse these using the
WebCam.)
- A
gzipped file containing just the sequences and their A-numbers (about 9 megs)
- A
gzipped file containing just the names of the sequences and their A-numbers (about 3 megs)
- Contributing a new sequence
(or a comment on
an existing sequence, or more terms for an existing sequence).
Want to help?
Set the
WebCam
to browse the sequences that need extending,
or use the
main look-up page
to search for keyword:more.
See also the
future projects
web page.
Other related pages:
Demos,
Transformations of sequences,
Maple or
Mathematica scripts to format sequences.
-
Sequences in Classic Books.
Comtet's Advanced Combinatorics,
Graham, Knuth and Patashnik's Concrete Mathematics,
Harary and Palmer's Graphical Enumeration,
Stanley's Enumerative Combinatorics.
-
Papers Citing
the Encyclopedia of Integer Sequences.
Shows
some of the ways that people have used the database.
- Referencing the OEIS. If the database helped
your work and you wish to reference it, the usual citation is something like this:
N. J. A. Sloane, (2007), The On-Line Encyclopedia of Integer Sequences,
www.research.att.com/~njas/sequences/.
Or, since that often causes spacing problems with LaTeX (the line is too long
and is hard to break):
N. J. A. Sloane, (2007), The On-Line Encyclopedia of Integer Sequences,
published electronically at www.research.att.com/~njas/sequences/.
Another possibility is to say something like:
N. J. A. Sloane,
The On-Line Encyclopedia of Integer Sequences.
World-Wide Web URL www.research.att.com/~njas/sequences/.
- URLs
The URL for the main lookup page is
http://www.research.att.com/~njas/sequences/
(or
http://www.research.att.com/%7enjas/sequences/
if your keyboard lacks the tilde character).
The URL for this page is
http://www.research.att.com/~njas/sequences/Seis.html
(or
http://www.research.att.com/%7enjas/sequences/Seis.html
if your keyboard lacks the tilde character).
- Referencing a Particular Sequence. If you are writing a paper
and wish to refer the Catalan numbers, say
(sequence
A000108),
but don't want to
digress to describe them, simply add a link that points directly to that sequence in the
database.
The URL for sequence A000108 (for example) is
http://www.research.att.com/~njas/sequences/A000108
(this is new short URL introduced Jan 01 27, 2006).
In an HTML file one might say something like this:
... where the C(n) are the Catalan numbers
(<a href="http://www.research.att.com/~njas/sequences/A000108">Sequence A000108</a> in [OEIS]).
One can also create active links in PDF or POSTSCRIPT files. From LATEX for example one can
use the
HYPERREF
package. In that case one would say:
... where the C(n) are the Catalan numbers
(sequence \htmladdnormallink{A000108}
{http://www.research.att.com/projects/OEIS?Anum=000108} in \cite{OEIS}).
For an example of a LATEX file which produces active links in this way,
see
"My Favorite Integer Sequences"
in three versions:
LATEX,
PDF and
POSTSCRIPT.
[That LATEX file uses the old style of links to the OEIS,
and needs to be changed.]
- URL for Searching the Database
To bypass the web page and search for a sequence directly using
the cgi program, for instance the sequence 2,5,14,50,233,
use (with no line break and no
internal spaces):
http://www.research.att.com/~njas/sequences/?q=2,5,14,50,233
To put a window on your own page to do lookups, use the following html commands:
To look up a number sequence in the
<a href="http://www.research.att.com/~njas/sequences/">
On-Line Encyclopedia of Integer Sequences</a>,
enter it here and click "Submit":
<form
action="http://www.research.att.com/~njas/sequences/"
method=get>
<input type=text name=q SIZE=60 VALUE=
"1,2,3,6,11,23,47,106,235">
<input type=submit VALUE="Submit">
</form>
To bypass the web page and search for a word or phrase directly using
the cgi program, for instance the phrase "number of factors",
use (with no line break and no
internal spaces):
http://www.research.att.com/~njas/sequences/?q="number of factors"
- Policy on Email Addresses in the OEIS
If possible I prefer to give the author's name and email address
with each sequence, so that people can get in touch with each other.
This is an important feature of the database.
Email addresses are disguised by replacing @ by (AT).
Let me know if you don't want your email address to
appear in any form.
However, if you ask to have your email address removed,
try to give me a link to your home page
- send me a line that looks like this:
%H A077001 John Smith, <a href="http://members.aol.org/~JSmth/">Home Page</a> (listed instead of email address)
that I can add to each sequence.
Again, when sending in a sequence or comment using the
Contribute new seq. or comment
web page,
if you don't want your email address to be used, say so
in one of the windows,
and if possible put the URL of your home page into one of the "links" windows.
- Copyright Notice. This database and its associated files are copyright
1996-2006 by N. J. A. Sloane.
- Acknowledgments. A very large number of people have
contributed to the table, and it is impossible to thank them individually.
Their names can be seen in the "Author" and "Extension" lines of the entries.
The following are some of the people who
have made major contributions in recent years.
Antonio G. Astudillo (afg_astudillo(AT)hotmail.com),
Asher Natan Auel (auela(AT)reed.edu),
Lekraj Beedassy (beedassylekraj(AT)hotmail.com),
Mira Bernstein (mira(AT)math.Stanford.edu).
Henry Bottomley,
Christian Bower (bowerc(AT)usa.net),
Benoit Cloitre (abcloitre(AT)wanadoo.fr),
John Conway (conway(AT)math.princeton.edu),
Patrick De Geest,
Patrick Demichel,
Frank Ellermann,
Steven Finch,
Erich Friedman,
Olivier Gerard,
Richard K. Guy (rkg(AT)cpsc.ucalgary.ca),
Vladeta Jovovic (vladeta(AT)Eunet.yu),
Clark Kimberling,
Elemer Labos (labos(AT)ana1.sote.hu),
Wolfdieter Lang,
Amarnath Murthy (amarnath_murthy(AT)yahoo.com),
T. D. Noe (noe(AT)sspectra.com), who has provided
extended version ("b-files") for nearly 3000 sequences,
Simon Plouffe,
Larry Reeves (larryr(AT)acm.org),
Francisco de Salinas,
James Sellers,
Jeffrey Shallit,
Michael Somos,
Ralf Stephan (ralf(AT)ark.in-berlin.de),
Eric Weisstein,
Barry E. Williams,
David W. Wilson (davidwwilson(AT)attbi.com),
Robert G. Wilson V (rgwv(AT)rgwv.com)
and
Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com).
Special thanks to Antti Karttunen,
who wrote the program that
displays sequences based on arrays (those with keyword "tabl")
in three different two-dimensional formats.
To see this, look at some of the following sequences,
and click on the keyword "tabl":
At the end of 2005
Alex Healy
and
Russ Cox (rsc(AT)swtch.com)
made a huge contribution to OEIS by greatly speeding up
the search process.
The first versions of the new programs were written
by Alex Healy and the final versions by Russ Cox.
My colleague David Applegate then helped install them
on our new server.
The new searches are much faster than the old ones
and can handle much more complicated queries.
See the
hints file
for details.
- Links.
-
Awards, etc.
NJAS's 2003 article
about the OEIS received the
Math. Assoc. America David R. Robbins Prize, Jan 07, 2008.
The OEIS was mentioned on the TV program
Numb3rs, May 05 2006.
Featured in Science News Online, May 17, 2003.
Written up in the
Frankfurter
Allgemeine Zeitung on May 9, 2001, and by
Slashdot
on Feb 22, 2000.
One of Science magazine's Hot Picks for 15 May 1998.
The email servers
were written up in Newsweek's "Cyberscope" column
on Jan. 9, 1995;
in Science on July 22, 1994; and in several
other places.
Also:
(Dec 05, 2007)
|
(May 10, 2001)
|
(Mar 09, 2000)
|
|
(Jun 12, 2000)
|
|
(May 15, 1998)
|
(Apr 29, 1997)
|
|
(1997)
|
|
(May 22, 1997)
|
|
(Oct. 9, 1996)
|
|
(1995)
|
|
Click the single right arrow to go to the next demonstration page,
or the single left arrow to return to the previous page.
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