On-Line Encyclopedia of Integer Sequences
Encyclopedia
The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane's, is an online database of integer sequence
s, created and maintained by N. J. A. Sloane
, a researcher at AT&T Labs
. In October 2009, the intellectual property
and hosting of the OEIS were transferred to the OEIS Foundation.
OEIS records information on integer sequences of interest to both professional mathematician
s and amateurs
, and is widely cited. it contains over 200,000 sequences, making it the largest database of its kind.
Each entry contains the leading terms of the sequence, keyword
s, mathematical motivations, literature links, and more, including the option to generate a graph
or play a musical
representation of the sequence. The database is searchable by keyword and by subsequence.
started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics
. The database was at first stored on punch card
s. He published selections from the database in book form twice:
These books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service (August 1994), and soon after as a web site (1996). As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998.
The database continues to grow at a rate of some 10,000 entries a year.
Sloane has personally managed 'his' sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database.
In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, , which counts the marks on the Ishango bone
. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS.org was created to simplify the collaboration of the OEIS editors and contributors. The 200,000th sequence, , was added to the database in November 2011; it was initially entered as A200715, and moved to A200000 after a week of discussion on the SeqFan mailing list, following a proposal by OEIS Editor-in-Chief Charles Greathouse to choose a a special sequence for A200000.
The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane's, is an online database of integer sequence
s, created and maintained by N. J. A. Sloane
, a researcher at AT&T Labs
. In October 2009, the intellectual property
and hosting of the OEIS were transferred to the OEIS Foundation.Transfer of IP in OEIS to The OEIS Foundation Inc.
OEIS records information on integer sequences of interest to both professional mathematician
s and amateurs
, and is widely cited. it contains over 200,000 sequences, making it the largest database of its kind.
Each entry contains the leading terms of the sequence, keyword
s, mathematical motivations, literature links, and more, including the option to generate a graph
or play a musical
representation of the sequence. The database is searchable by keyword and by subsequence.
started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics
. The database was at first stored on punch card
s. He published selections from the database in book form twice:
These books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service (August 1994), and soon after as a web site (1996). As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998.Journal of Integer Sequences
The database continues to grow at a rate of some 10,000 entries a year.
Sloane has personally managed 'his' sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database.
In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, , which counts the marks on the Ishango bone
. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS.org was created to simplify the collaboration of the OEIS editors and contributors. The 200,000th sequence, , was added to the database in November 2011; it was initially entered as A200715, and moved to A200000 after a week of discussion on the SeqFan mailing list, following a proposal by OEIS Editor-in-Chief Charles Greathouse to choose a a special sequence for A200000.
The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane's, is an online database of integer sequence
s, created and maintained by N. J. A. Sloane
, a researcher at AT&T Labs
. In October 2009, the intellectual property
and hosting of the OEIS were transferred to the OEIS Foundation.Transfer of IP in OEIS to The OEIS Foundation Inc.
OEIS records information on integer sequences of interest to both professional mathematician
s and amateurs
, and is widely cited. it contains over 200,000 sequences, making it the largest database of its kind.
Each entry contains the leading terms of the sequence, keyword
s, mathematical motivations, literature links, and more, including the option to generate a graph
or play a musical
representation of the sequence. The database is searchable by keyword and by subsequence.
started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics
. The database was at first stored on punch card
s. He published selections from the database in book form twice:
These books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service (August 1994), and soon after as a web site (1996). As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998.Journal of Integer Sequences
The database continues to grow at a rate of some 10,000 entries a year.
Sloane has personally managed 'his' sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database.
In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, , which counts the marks on the Ishango bone
. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS.org was created to simplify the collaboration of the OEIS editors and contributors. The 200,000th sequence, , was added to the database in November 2011; it was initially entered as A200715, and moved to A200000 after a week of discussion on the SeqFan mailing list, following a proposal by OEIS Editor-in-Chief Charles Greathouse to choose a a special sequence for A200000.
s, the digits of transcendental number
s, complex number
s and so on by transforming them into integer sequences.
Sequences of rationals are represented by two sequences (named with the keyword 'frac'): the sequence of numerators and the sequence of denominators. For example, the fifth order Farey sequence
,
, is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5 .
Important irrational numbers such as π = 3.1415926535897... are catalogued under representative integer sequences such as decimal expansions (here 3, 1, 4, 1, 5, 9, 2, 6, ... ) or continued fraction
expansions (here 3, 7, 15, 1, 292, 1, ... ).
text, so it uses a linear form of conventional mathematical notation (such as f(n) for functions, n for running variables, etc.). Greek letters
are usually represented by their full names, e.g., mu for μ, phi for φ.
Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, e.g., A315 rather than A000315.
Individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces.
In comments, formulas, etc., a(n) represents the nth term of the sequence.
of least magic constant, or 0 if no such magic square exists." The value of a(1) (a 1×1 magic square) is 2; a(3) is 1480028129. But there is no such 2×2 magic square, so a(2) is 0.
This special usage has a solid mathematical basis in certain counting functions. For example, the totient valence function
counts the solutions of φ(x) = m. There are 4 solutions for 4, but no solutions for 14, hence a(14) of A014197 is 0—there are no solutions.
Occasionally -1 is used for this purpose instead, as in .
of the sequences, so each sequence has a predecessor and a successor (its "context"). OEIS normalizes the sequences for lexicographical ordering, (usually) ignoring initial zeros or ones and also the sign of each element. Sequences of weight distribution
codes often omit periodically recurring zeros.
For example, consider: the prime number
s, the palindromic prime
s, the Fibonacci sequence
, the lazy caterer's sequence
, and the coefficients in the series expansion of
. In OEIS lexicographic order, they are:
whereas unnormalized lexicographic ordering would order these sequences thus: #3, #5, #4, #1, #2.
One of the earliest self-referential sequences Sloane accepted into the OEIS was (later ) "a(n) = n-th term of sequence An". This sequence spurred progress on finding more terms of . Some sequences are both finite and listed in full (keywords "fini" and "full"); these sequences will not always be long enough to contain a term that corresponds to their OEIS sequence number. In this case the corresponding term a(n) of A091967 is undefined.
lists the first term given in sequence An, but it needs to be updated from time to time because of changing opinions on offsets. Listing instead term a(1) of sequence An might seem a good alternative if it weren't for the fact that some sequences have offsets of 2 and greater.
This line of thought leads to the question "Does sequence An contain the number n ?" and the sequences , "Numbers n such that OEIS sequence An contains n", and , "n is in this sequence if and only if n is not in sequence An". Thus, the composite number 2808 is in A053873 because is the sequence of composite numbers, while the non-prime 40 is in A053169 because it's not in A000040, the prime numbers. Each n is a member of exactly one of these two sequences, and in principle it can be determined which sequence each n belongs to, with two exceptions (related to the two sequences themselves):
program, it contains every field an OEIS entry can have.
ID number
Sequence data
Name
Comments
References
Links
Formula
Example
Maple
Mathematica
Program
See also
Keyword
Offset
Author(s)
Extension
You can enter negative signs, but they will be ignored. For example, 0, 3, 7, 13, 20, 28, 36, 43, 47, 45, 32, 0, −64, n2 minus the nth Fibonacci number, is a sequence that is technically not in the OEIS, but the very similar sequence 0, −3, −7, −13, −20, −28, −36, −43, −47, −45, −32, 0, 64, is in the OEIS and will come up when one searches for its reversed signs counterpart.
However, the search can be forced to match signs by using the prefix "signed:" in the search string. This is especially useful for sequences like that consist exclusively of positive and negative ones.
One can enter as little as a single integer or as much as four lines of terms. Sloane recommends entering six terms, a(2) to a(7), in order to get enough results, but not too many results. There are cases where entering just one integer gives precisely one result, such as 6610199 brings up just , the strobogrammatic prime
s which are not palindromic). There are also cases where one can enter many terms and still not narrow the results down very much.
, try enter it without the accents: "Znam's problem." The handling of apostrophes has been greatly improved in the 2006 redesign. The search strings "Pascal's triangle
," "Pascals triangle" and "Pascal triangle" all give the desired results.
To look up most polygonal number
s by word, try "n-gonal numbers" rather than "Greek prefix-gonal numbers" (e.g., "47-gonal numbers" instead of "heptaquartagonal numbers"). Beyond "dodecagonal numbers," word searching with the Greek prefixes might fail to yield the desired results.
Integer sequence
In mathematics, an integer sequence is a sequence of integers.An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms...
s, created and maintained by N. J. A. Sloane
Neil Sloane
Neil James Alexander Sloane is a British-U.S. mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, and sphere packing...
, a researcher at AT&T Labs
AT&T Labs
AT&T Labs, Inc. is the research & development division of AT&T, where scientists and engineers work to understand and advance innovative technologies relevant to networking, communications, and information. Over 1800 employees work in six locations: Florham Park, NJ; Middletown, NJ; Austin, TX;...
. In October 2009, the intellectual property
Intellectual property
Intellectual property is a term referring to a number of distinct types of creations of the mind for which a set of exclusive rights are recognized—and the corresponding fields of law...
and hosting of the OEIS were transferred to the OEIS Foundation.
OEIS records information on integer sequences of interest to both professional mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s and amateurs
Recreational mathematics
Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...
, and is widely cited. it contains over 200,000 sequences, making it the largest database of its kind.
Each entry contains the leading terms of the sequence, keyword
Keyword (computer programming)
In computer programming, a keyword is a word or identifier that has a particular meaning to the programming language. The meaning of keywords — and, indeed, the meaning of the notion of keyword — differs widely from language to language....
s, mathematical motivations, literature links, and more, including the option to generate a graph
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...
or play a musical
Computer music
Computer music is a term that was originally used within academia to describe a field of study relating to the applications of computing technology in music composition; particularly that stemming from the Western art music tradition...
representation of the sequence. The database is searchable by keyword and by subsequence.
History
Neil SloaneNeil Sloane
Neil James Alexander Sloane is a British-U.S. mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, and sphere packing...
started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
. The database was at first stored on punch card
Punch card
A punched card, punch card, IBM card, or Hollerith card is a piece of stiff paper that contains digital information represented by the presence or absence of holes in predefined positions...
s. He published selections from the database in book form twice:
- A Handbook of Integer Sequences (1973, ISBN 0-12-648550-X), containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372.
- The Encyclopedia of Integer Sequences with Simon PlouffeSimon PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...
(1995, ISBN 0-12-558630-2), containing 5,488 sequences and assigned M-numbers from M0000 to M5487. The Encyclopedia includes the references to the corresponding sequences (which may differ in their few initial terms) in A Handbook of Integer Sequences as N-numbers from N0001 to N2372 (instead of 1 to 2372.) The Encyclopedia includes the A-numbers that are used in the OEIS, whereas the Handbook did not.
These books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service (August 1994), and soon after as a web site (1996). As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998.
The database continues to grow at a rate of some 10,000 entries a year.
Sloane has personally managed 'his' sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database.
In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, , which counts the marks on the Ishango bone
Ishango bone
The Ishango bone is a bone tool, dated to the Upper Paleolithic era. It is a dark brown length of bone, the fibula of a baboon, with a sharp piece of quartz affixed to one end, perhaps for engraving...
. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS.org was created to simplify the collaboration of the OEIS editors and contributors. The 200,000th sequence, , was added to the database in November 2011; it was initially entered as A200715, and moved to A200000 after a week of discussion on the SeqFan mailing list, following a proposal by OEIS Editor-in-Chief Charles Greathouse to choose a a special sequence for A200000.
The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane's, is an online database of integer sequence
Integer sequence
In mathematics, an integer sequence is a sequence of integers.An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms...
s, created and maintained by N. J. A. Sloane
Neil Sloane
Neil James Alexander Sloane is a British-U.S. mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, and sphere packing...
, a researcher at AT&T Labs
AT&T Labs
AT&T Labs, Inc. is the research & development division of AT&T, where scientists and engineers work to understand and advance innovative technologies relevant to networking, communications, and information. Over 1800 employees work in six locations: Florham Park, NJ; Middletown, NJ; Austin, TX;...
. In October 2009, the intellectual property
Intellectual property
Intellectual property is a term referring to a number of distinct types of creations of the mind for which a set of exclusive rights are recognized—and the corresponding fields of law...
and hosting of the OEIS were transferred to the OEIS Foundation.Transfer of IP in OEIS to The OEIS Foundation Inc.
OEIS records information on integer sequences of interest to both professional mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s and amateurs
Recreational mathematics
Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...
, and is widely cited. it contains over 200,000 sequences, making it the largest database of its kind.
Each entry contains the leading terms of the sequence, keyword
Keyword (computer programming)
In computer programming, a keyword is a word or identifier that has a particular meaning to the programming language. The meaning of keywords — and, indeed, the meaning of the notion of keyword — differs widely from language to language....
s, mathematical motivations, literature links, and more, including the option to generate a graph
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...
or play a musical
Computer music
Computer music is a term that was originally used within academia to describe a field of study relating to the applications of computing technology in music composition; particularly that stemming from the Western art music tradition...
representation of the sequence. The database is searchable by keyword and by subsequence.
History
Neil SloaneNeil Sloane
Neil James Alexander Sloane is a British-U.S. mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, and sphere packing...
started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
. The database was at first stored on punch card
Punch card
A punched card, punch card, IBM card, or Hollerith card is a piece of stiff paper that contains digital information represented by the presence or absence of holes in predefined positions...
s. He published selections from the database in book form twice:
- A Handbook of Integer Sequences (1973, ISBN 0-12-648550-X), containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372.
- The Encyclopedia of Integer Sequences with Simon PlouffeSimon PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...
(1995, ISBN 0-12-558630-2), containing 5,488 sequences and assigned M-numbers from M0000 to M5487. The Encyclopedia includes the references to the corresponding sequences (which may differ in their few initial terms) in A Handbook of Integer Sequences as N-numbers from N0001 to N2372 (instead of 1 to 2372.) The Encyclopedia includes the A-numbers that are used in the OEIS, whereas the Handbook did not.
These books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service (August 1994), and soon after as a web site (1996). As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998.Journal of Integer Sequences
The database continues to grow at a rate of some 10,000 entries a year.
Sloane has personally managed 'his' sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database.
In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, , which counts the marks on the Ishango bone
Ishango bone
The Ishango bone is a bone tool, dated to the Upper Paleolithic era. It is a dark brown length of bone, the fibula of a baboon, with a sharp piece of quartz affixed to one end, perhaps for engraving...
. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS.org was created to simplify the collaboration of the OEIS editors and contributors. The 200,000th sequence, , was added to the database in November 2011; it was initially entered as A200715, and moved to A200000 after a week of discussion on the SeqFan mailing list, following a proposal by OEIS Editor-in-Chief Charles Greathouse to choose a a special sequence for A200000.
The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane's, is an online database of integer sequence
Integer sequence
In mathematics, an integer sequence is a sequence of integers.An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms...
s, created and maintained by N. J. A. Sloane
Neil Sloane
Neil James Alexander Sloane is a British-U.S. mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, and sphere packing...
, a researcher at AT&T Labs
AT&T Labs
AT&T Labs, Inc. is the research & development division of AT&T, where scientists and engineers work to understand and advance innovative technologies relevant to networking, communications, and information. Over 1800 employees work in six locations: Florham Park, NJ; Middletown, NJ; Austin, TX;...
. In October 2009, the intellectual property
Intellectual property
Intellectual property is a term referring to a number of distinct types of creations of the mind for which a set of exclusive rights are recognized—and the corresponding fields of law...
and hosting of the OEIS were transferred to the OEIS Foundation.Transfer of IP in OEIS to The OEIS Foundation Inc.
OEIS records information on integer sequences of interest to both professional mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s and amateurs
Recreational mathematics
Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...
, and is widely cited. it contains over 200,000 sequences, making it the largest database of its kind.
Each entry contains the leading terms of the sequence, keyword
Keyword (computer programming)
In computer programming, a keyword is a word or identifier that has a particular meaning to the programming language. The meaning of keywords — and, indeed, the meaning of the notion of keyword — differs widely from language to language....
s, mathematical motivations, literature links, and more, including the option to generate a graph
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...
or play a musical
Computer music
Computer music is a term that was originally used within academia to describe a field of study relating to the applications of computing technology in music composition; particularly that stemming from the Western art music tradition...
representation of the sequence. The database is searchable by keyword and by subsequence.
History
Neil SloaneNeil Sloane
Neil James Alexander Sloane is a British-U.S. mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, and sphere packing...
started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
. The database was at first stored on punch card
Punch card
A punched card, punch card, IBM card, or Hollerith card is a piece of stiff paper that contains digital information represented by the presence or absence of holes in predefined positions...
s. He published selections from the database in book form twice:
- A Handbook of Integer Sequences (1973, ISBN 0-12-648550-X), containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372.
- The Encyclopedia of Integer Sequences with Simon PlouffeSimon PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...
(1995, ISBN 0-12-558630-2), containing 5,488 sequences and assigned M-numbers from M0000 to M5487. The Encyclopedia includes the references to the corresponding sequences (which may differ in their few initial terms) in A Handbook of Integer Sequences as N-numbers from N0001 to N2372 (instead of 1 to 2372.) The Encyclopedia includes the A-numbers that are used in the OEIS, whereas the Handbook did not.
These books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service (August 1994), and soon after as a web site (1996). As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998.Journal of Integer Sequences
The database continues to grow at a rate of some 10,000 entries a year.
Sloane has personally managed 'his' sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database.
In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, , which counts the marks on the Ishango bone
Ishango bone
The Ishango bone is a bone tool, dated to the Upper Paleolithic era. It is a dark brown length of bone, the fibula of a baboon, with a sharp piece of quartz affixed to one end, perhaps for engraving...
. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS.org was created to simplify the collaboration of the OEIS editors and contributors. The 200,000th sequence, , was added to the database in November 2011; it was initially entered as A200715, and moved to A200000 after a week of discussion on the SeqFan mailing list, following a proposal by OEIS Editor-in-Chief Charles Greathouse to choose a a special sequence for A200000.
Non-integers
Besides integer sequences, the OEIS also catalogs sequences of fractionFraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
s, the digits of transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
s, complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s and so on by transforming them into integer sequences.
Sequences of rationals are represented by two sequences (named with the keyword 'frac'): the sequence of numerators and the sequence of denominators. For example, the fifth order Farey sequence
Farey sequence
In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size....
,
, is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5 .Important irrational numbers such as π = 3.1415926535897... are catalogued under representative integer sequences such as decimal expansions (here 3, 1, 4, 1, 5, 9, 2, 6, ... ) or continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
expansions (here 3, 7, 15, 1, 292, 1, ... ).
Conventions
The OEIS is currently limited to plain ASCIIASCII
The American Standard Code for Information Interchange is a character-encoding scheme based on the ordering of the English alphabet. ASCII codes represent text in computers, communications equipment, and other devices that use text...
text, so it uses a linear form of conventional mathematical notation (such as f(n) for functions, n for running variables, etc.). Greek letters
Greek alphabet
The Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...
are usually represented by their full names, e.g., mu for μ, phi for φ.
Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, e.g., A315 rather than A000315.
Individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces.
In comments, formulas, etc., a(n) represents the nth term of the sequence.
Special meaning of zero
Zero is often used to represent non-existent sequence elements. For example, enumerates the "smallest prime of n² consecutive primes to form an n×n magic squareMagic square
In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...
of least magic constant, or 0 if no such magic square exists." The value of a(1) (a 1×1 magic square) is 2; a(3) is 1480028129. But there is no such 2×2 magic square, so a(2) is 0.
This special usage has a solid mathematical basis in certain counting functions. For example, the totient valence function
counts the solutions of φ(x) = m. There are 4 solutions for 4, but no solutions for 14, hence a(14) of A014197 is 0—there are no solutions.Occasionally -1 is used for this purpose instead, as in .
Lexicographical ordering
The OEIS maintains the lexicographical orderLexicographical order
In mathematics, the lexicographic or lexicographical order, , is a generalization of the way the alphabetical order of words is based on the alphabetical order of letters.-Definition:Given two partially ordered sets A and B, the lexicographical order on...
of the sequences, so each sequence has a predecessor and a successor (its "context"). OEIS normalizes the sequences for lexicographical ordering, (usually) ignoring initial zeros or ones and also the sign of each element. Sequences of weight distribution
Weight distribution
Weight distribution is the apportioning of weight within a vehicle, especially cars, airplanes, and trains.In a vehicle which relies on gravity in some way, weight distribution directly affects a variety of vehicle characteristics, including handling, acceleration, traction, and component life...
codes often omit periodically recurring zeros.
For example, consider: the prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s, the palindromic prime
Palindromic prime
A palindromic prime is a prime number that is also a palindromic number. Palindromicity depends on the base of the numbering system and its writing conventions, while primality is independent of such concerns...
s, the Fibonacci sequence
Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
, the lazy caterer's sequence
Lazy caterer's sequence
The lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a circle that can be made with a given number of straight cuts. For example, three cuts across a pancake will produce six pieces if the cuts all meet at a common point, but...
, and the coefficients in the series expansion of
. In OEIS lexicographic order, they are:
- Sequence #1: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, ...
- Sequence #2: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, ...
- Sequence #3: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...
- Sequence #4: 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, ...
- Sequence #5: 1, −3, −8, −3, −24, 24, −48, −3, −8, 72, −120, 24, −168, 144, ...
whereas unnormalized lexicographic ordering would order these sequences thus: #3, #5, #4, #1, #2.
Self-referential sequences
Very early in the history of the OEIS, sequences defined in terms of the numbering of sequences in the OEIS itself were proposed. "I resisted adding these sequences for a long time, partly out of a desire to maintain the dignity of the database, and partly because A22 was only known to 11 terms !" Sloane reminisced.One of the earliest self-referential sequences Sloane accepted into the OEIS was (later ) "a(n) = n-th term of sequence An". This sequence spurred progress on finding more terms of . Some sequences are both finite and listed in full (keywords "fini" and "full"); these sequences will not always be long enough to contain a term that corresponds to their OEIS sequence number. In this case the corresponding term a(n) of A091967 is undefined.
lists the first term given in sequence An, but it needs to be updated from time to time because of changing opinions on offsets. Listing instead term a(1) of sequence An might seem a good alternative if it weren't for the fact that some sequences have offsets of 2 and greater.
This line of thought leads to the question "Does sequence An contain the number n ?" and the sequences , "Numbers n such that OEIS sequence An contains n", and , "n is in this sequence if and only if n is not in sequence An". Thus, the composite number 2808 is in A053873 because is the sequence of composite numbers, while the non-prime 40 is in A053169 because it's not in A000040, the prime numbers. Each n is a member of exactly one of these two sequences, and in principle it can be determined which sequence each n belongs to, with two exceptions (related to the two sequences themselves):
- It cannot be determined whether 53873 is a member of A053873 or not. If it is in the sequence then by definition it should be; if it is not in the sequence then (again, by definition) it should not be.
- It can be proved that 53169 both is and is not a member of A053169. If it is in the sequence then it should not be; if it is not in the sequence then it should be. This is a form of Russell's paradoxRussell's paradoxIn the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
.
An abridged example of a typical OEIS entry
This entry, , was chosen because, with the exception of a MapleMaple (software)
Maple is a general-purpose commercial computer algebra system. It was first developed in 1980 by the Symbolic Computation Group at the University of Waterloo in Waterloo, Ontario, Canada....
program, it contains every field an OEIS entry can have.
A046970 Generated from Riemann Zeta function: coefficients in series expansion of Zeta(n+2)/Zeta(n).
1, -3, -8, -3, -24, 24, -48, -3, -8, 72, -120, 24, -168, 144, 192, -3, -288, 24, -360, 72, 384, 360, -528, 24, -24, 504, -8, 144, -840, -576, -960, -3, 960, 864, 1152, 24, -1368, 1080, 1344, 72, -1680, -1152, -1848, 360, 192, 1584, -2208, 24, -48, 72, 2304, 504, -2808, 24, 2880, 144, 2880, 2520, -3480, -576
OFFSET 1,2
COMMENTS B(n+2) = -B(n)*((n+2)*(n+1)/(4pi^2))*z(n+2)/z(n) = -B(n)*((n+2)*(n+1)/(4pi^2))*Sum(j=1, infinity) [ a(j)/j^(n+2) ]
...
REFERENCES M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, pp. 805-811.
LINKS M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Wikipedia, Riemann zeta function.
FORMULA Multiplicative with a(p^e) = 1-p^2. a(n) = Sum_{d|n} mu(d)*d^2.
a(n) = product[p prime divides n, p^2-1] (gives unsigned version) [From Jon Perry (jonperrydc(AT)btinternet.com), Aug 24 2010]
EXAMPLE a(3) = -8 because the divisors of 3 are {1, 3} and mu(1)*1^2 + mu(3)*3^2 = -8.
...
MATHEMATICA muDD[d_] := MoebiusMu[d]*d^2; Table[Plus @@ muDD[Divisors[n]], {n, 60}] (Lopez)
Flatten[Table[{ x = FactorInteger[n]; p = 1; For[i = 1, i <= Length[x], i++, p = p*(xiII is the ninth letter and a vowel in the basic modern Latin alphabet.-History:In Semitic, the letter may have originated in a hieroglyph for an arm that represented a voiced pharyngeal fricative in Egyptian, but was reassigned to by Semites, because their word for "arm" began with that sound...
11Year 1 was a common year starting on Saturday or Sunday of the Julian calendar and a common year starting on Saturday of the Proleptic Julian calendar...
^2 - 1)]; p}, {n, 1, 50, 1}]] [From Jon Perry (jonperrydc(AT)btinternet.com), Aug 24 2010]
PROG (PARI) A046970(n)=sumdiv(n, d, d^2*moebius(d)) (Benoit Cloitre)
CROSSREFS Cf. A027641 and A027642.
Sequence in context: A035292 A144457 A146975 * A058936 A002017 A118582
Adjacent sequences: A046967 A046968 A046969 * A046971 A046972 A046973
KEYWORD sign,mult
AUTHOR Douglas Stoll, dougstoll(AT)email.msn.com
EXTENSIONS Corrected and extended by Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 25 2001
Additional comments from Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Jul 01 2005
Entry fields
See Format of OEIS Pages.ID number
- Every sequence in the OEIS has a serial numberSerial numberA serial number is a unique number assigned for identification which varies from its successor or predecessor by a fixed discrete integer value...
, a six-digit positive integer, prefixed by A (and zero-padded on the left prior to November 2004). The letter "A" stands for "absolute." Numbers are either assigned by the editor(s) or by an A number dispenser, which is handy for when contributors wish to send in related sequences at once and be able to create cross-references. An A number from the dispenser expires a month from issue if not used. But as the following table of arbitrarily selected sequences show, the rough correspondence holds.
| Numbers n such that the binomial coefficient C(2n, n) is not divisible by the square of an odd prime. | January 1, 2001 | |
| Fibonacci Fibonacci number In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; .... (n)!. |
March 14, 2001 | |
| Number of 3-dimensional polyominoes (or polycubes) with n cells and symmetry group of order exactly 24. | January 1, 2002 | |
| Smallest number such that n·a(n) is a concatenation of n consecutive integers ... | August 31, 2002 | |
| Continued fraction for ζ(3/2) | January 1, 2003 | |
| Number of permutations satisfying −k ≤ p(i) − i ≤ r and p(i) − i | February 10, 2003 | |
| Length of longest contiguous block of 1s in binary expansion of nth prime. | November 20, 2003 | |
| Exponential convolution of A069321(n) with itself, where we set A069321(0) = 0. | January 1, 2004 | |
| Marks from the 22000-year-old Ishango bone Ishango bone The Ishango bone is a bone tool, dated to the Upper Paleolithic era. It is a dark brown length of bone, the fibula of a baboon, with a sharp piece of quartz affixed to one end, perhaps for engraving... from the Congo. |
November 7, 2004 | |
| Column 1 of triangle A102230, and equals the convolution of A032349 with A032349 shift right. | January 1, 2005 | |
| Number of consecutive integers starting with n needed to sum to a Niven number. | July 8, 2005 | |
| Triangle-free positive integers. | January 12, 2006 | |
| Möbius transform of sum of prime factors of n. | June 2, 2006 |
- Even for sequences in the book predecessors to the OEIS, the ID numbers are not the same. The 1973 Handbook of Integer Sequences contained about 2400 sequences, which were numbered by lexicographic order (the letter M plus 4 digits, zero-padded where necessary), and the 1995 Encyclopedia of Integer Sequences contained 5487 sequences, also numbered by lexicographic order (the letter N plus 4 digits, zero-padded where necessary). These old M and N numbers, as applicable, are contained in the ID number field in parentheses after the modern A number.
Sequence data
- The sequence field lists the numbers themselves, or at least about four lines' worth. The sequence field makes no distinction between sequences that are finite but still too long to display and sequences that are infinite. To help make that determination, you need to look at the keywords field for "fini," "full," or "more." To determine to which n the values given correspond, see the offset field, which gives the n for the first term given.
Name
- The name field usually contains the most common name for the sequence, and sometimes also the formula. For example, 1, 8, 27, 64, 125, 216, 343, 512, is named "The cubes: a(n) = n^3."
Comments
- The comments field is for information about the sequence that doesn't quite fit in any of the other fields. The comments field often points out interesting relationships between different sequences and less obvious applications for a sequence. For example, Lekraj Beedassy in a comment to A000578 notes that the cube numbers also count the "total number of triangles resulting from criss-crossing ceviansCeva's theoremCeva's theorem is a theorem about triangles in plane geometry. Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O to meet opposite sides at D, E and F respectively...
within a triangle so that two of its sides are each n-partitioned," while Sloane points out the unexpected relationship between centered hexagonal numbers and second Bessel polynomials in a comment to A003215.
References
- References to printed documents (books, papers, ...)
Links
- Links, i.e. URLUniform Resource LocatorIn computing, a uniform resource locator or universal resource locator is a specific character string that constitutes a reference to an Internet resource....
s, to online resources. These may be:- links to the index
- links to ASCII files which hold the sequence terms in a two column format over a wider range of indices than held by the main database lines
- links to images in the local database directories which often provide combinatorial background related to graph theory
- references to applicable articles in journals
- others related to computer codes, more extensive tabulations in specific research areas provided by indivduals or research groups
Formula
- Formulae, recurrences, generating functions, etc. for the sequence.
Example
- Some examples of sequence member values.
Maple
- Maple code.
Mathematica
- MathematicaMathematicaMathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...
code.
Program
- Maple and MathematicaMathematicaMathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...
are the preferred programs for calculating sequences in the OEIS, and they both get their own field labels, "Maple" and "Mathematica." As of Jan 2009, Mathematica is the most popular choice with over 25,000 Mathematica programs followed by 13,000 Maple programs. There are 11,000 programs in PARIPARI/GPPARI/GP is a computer algebra system with the main aim of facilitating number theory computations. It is free software; versions 2.1.0 and higher are distributed under the GNU General Public License...
and 3000 in other languages, all of which are labelled with a generic "Program" field label and the name of the program in parentheses. - If there is no name given, the program was written by the original submitter of the sequence.
See also
- Sequence cross-references originated by the original submitter are usually denoted by "Cf."
- Except for new sequences, the see also field also includes information on the lexicographic order of the sequence (its "context") and provides links to sequences with close A numbers (A046967, A046968, A046969, A046971, A046972, A046973, in our example). The following table shows the context of our example sequence, A046970:
| 3, 8, 3, 9, 4, 5, 2, 3, 1, 2, ... | Decimal expansion of ln(93/2). | |
| 1, 1, 1, 3, 8, 3, 10, 1, 110, 3, 406, 3 | First numerator and then denominator of the central elements of the 1/3-Pascal triangle (by row). |
|
| 1, 3, 8, 3, 12, 24, 16, 3, 41, 36, 24, ... | Number of similar sublattices of Z4 of index n2. | |
| 1, −3, −8, −3, −24, 24, −48, −3, −8, 72, ... | Generated from Riemann zeta function... | |
| 0, 1, 3, 8, 3, 30, 20, 144, 90, 40, 840, 504, 420, 5760, 3360, 2688, 1260 |
Decomposition of Stirling's S(n, 2) based on associated numeric partitions. |
|
| 1, 1, 1, 0, −3, −8, −3, 56, 217, 64, −2951, −12672, ... | Expansion of exp(sin x). | |
| 3, 8, 4, 1, 4, 9, 9, 0, 0, 7, 5, 4, 3, 5, 0, 7, 8 | Decimal expansion of upper bound for the r-values supporting stable period-3 orbits in the logistic equation. |
Keyword
- The OEIS has its own standard set of four or five letter keywords that characterize each sequence:
- base The results of the calculation depend on a specific positional base. For example, 2, 3, 5, 7, 11, 101, 131, 151, 181 ... are prime numbers regardless of base, but they are palindromicPalindromic primeA palindromic prime is a prime number that is also a palindromic number. Palindromicity depends on the base of the numbering system and its writing conventions, while primality is independent of such concerns...
specifically in base 10. Most of them are not palindromic in binary. Some sequences rate this keyword depending on how they're defined. For example, the Mersenne primeMersenne primeIn mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...
s 3, 7, 31, 127, 8191, 131071, ... does not rate "base" if defined as "primes of the form 2^n - 1." However, defined as "repunitRepunitIn recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler...
primes in binary," the sequence would rate the keyword "base." - bref "sequence is too short to do any analysis with", for example, , Number of isomorphism classes of associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n.
- cofr The sequence represents a continued fractionContinued fractionIn mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
. - cons The sequence is a decimal expansion of an important mathematical constant, like e or π.
- core A sequence that is of foundational importance to a branch of mathematics, such as the prime numbers, the Fibonacci sequence, etc.
- dead This keyword used for erroneous sequences that have appeared in papers or books, or for duplicates of existing sequences. For example, is the same as A000668.
- dumb One of the more subjective keywords, for "unimportant sequences," which may or may not directly relate to mathematics. , "Mix digits of pi and e." is one example of the former, and , "Numbers on a computer keyboard, read in a spiral." is an example of the latter.
- easy The terms of the sequence can be easily calculated. Perhaps the sequence most deserving of this keyword is 1, 2, 3, 4, 5, 6, 7, ... , where each term is 1 more than the previous term. The keyword "easy" is sometimes given to sequences "primes of the form f(m)" where f(m) is an easily calculated function. (Though even if f(m) is easy to calculate for large m, it might be very difficult to determine if f(m) is prime).
- eigen A sequence of eigenvalues.
- fini The sequence is finite, although it might still contain more terms than can be displayed. For example, the sequence field of shows only about a quarter of all the terms, but a comment notes that the last term is 3888.
- frac A sequence of either numerators or denominators of a sequence of fractions representing rational numbers. Any sequence with this keyword ought to be cross-referenced to its matching sequence of numerators or denominators, though this may be dispensed with for sequences of Egyptian fractions, such as , where the sequence of numerators would be . This keyword should not be used for sequences of continued fractions, cofr should be used instead for that purpose.
- full The sequence field displays the complete sequence. If a sequence has the keyword "full," it should also have the keyword "fini." One example of a finite sequence given in full is that of the supersingular primeSupersingular prime (moonshine theory)In the mathematical branch of moonshine theory, a supersingular prime is a certain type of prime number.Namely, a supersingular prime is a prime divisor of the order of the Monster group M, the largest of the sporadic simple groups...
s , of which there are precisely fifteen. - hard The terms of the sequence cannot be easily calculated, even with raw number crunching power. This keyword is most often used for sequences corresponding to unsolved problems, such as "How many n-spheres can touch another n-sphere of the same size?" lists the first ten known solutions.
- less A "less interesting sequence".
- more More terms of the sequence are wanted. Readers can submit an extension.
- mult The sequence corresponds to a multiplicative functionMultiplicative functionIn number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...
. Term a(1) should be 1, and term a(mn) can be calculated by multiplying a(m) by a(n) if m and n are coprime. For example, in , a(12) = a(3)a(4) = -8 × -3. - new For sequences that were added in the last couple of weeks, or had a major extension recently. This keyword is not given a checkbox in the Web form for submitting new sequences, Sloane's program adds it by default where applicable.
- nice Perhaps the most subjective keyword of all, for "exceptionally nice sequences."
- nonn The sequence consists of nonnegative integers (it may include zeroes). No distinction is made between sequences that consist of nonnegative numbers only because of the chosen offset (e.g., n3, the cubes, which are all positive from n = 0 forwards) and those that by definition are completely nonnegative (e.g., n2, the squares).
- obsc The sequence is considered obscure and needs a better definition.
- probation Sequences that "may be deleted later at the discretion of the editor."
- sign Some (or all) of the values of the sequence are negative. The entry includes both a Signed field with the signs and a Sequence field consisting of all the values passed through the absolute valueAbsolute valueIn mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
function. - tabf "An irregular (or funny-shaped) array of numbers made into a sequence by reading it row by row." For example, , "Triangle read by rows giving successive states of cellular automaton generated by "rule 62."
- tabl A sequence obtained by reading a geometric arrangement of numbers, such as a triangle or square, row by row. The quintessential example is Pascal's trianglePascal's triangleIn mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
read by rows, . - uned Sloane has not edited the sequence but believes it could be worth including in the OEIS. The sequence could contain computational or typographical errors. Contributors are invited to ponder the sequence and send Sloane their edition.
- unkn "Little is known" about the sequence, not even the formula that produces it. For example, , which was presented to the Internet OracleInternet OracleThe Internet Oracle is an effort at collective humor in a pseudo-Socratic question-and-answer format....
to ponder. - walk "Counts walks (or self-avoiding paths)."
- word Depends on the words of a specific language. For example, zero, one, two, three, four, five, etc., 4, 3, 3, 5, 4, 4, 3, 5, 5, 4, 3, 6, 6, 8, 8, 7, 7, 9, 8, 8 ... , "Number of letters in the English name of n, excluding spaces and hyphens."
- base The results of the calculation depend on a specific positional base. For example, 2, 3, 5, 7, 11, 101, 131, 151, 181 ... are prime numbers regardless of base, but they are palindromic
- Some keywords are mutually exclusive, namely: core and dumb, easy and hard, full and more, less and nice, and nonn and sign.
Offset
- The offset is the index of the first term given. For some sequences, the offset is obvious. For example, if we list the sequence of square numbers as 0, 1, 4, 9, 16, 25 ..., the offset is 0; while if we list it as 1, 4, 9, 16, 25 ..., the offset is 1. The default offset is 0, and most sequences in the OEIS have offset of either 0 or 1. Sequence , the magic constantMagic constantThe magic constant or magic sum of a magic square is the sum of numbers in any row, column, and diagonal of the magic square. For example, the magic square shown below has a magic constant of 15....
for n×n magic squareMagic squareIn recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...
with prime entries (regarding 1 as a prime) with smallest row sums, is an example of a sequence with offset 3, and , "Number of stars of visual magnitude n." is an example of a sequence with offset -1. Sometimes there can be disagreement over what the initial terms of the sequence are, and correspondingly what the offset should be. In the case of the lazy caterer's sequenceLazy caterer's sequenceThe lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a circle that can be made with a given number of straight cuts. For example, three cuts across a pancake will produce six pieces if the cuts all meet at a common point, but...
, the maximum number of pieces you can cut a pancake into with n cuts, the OEIS gives the sequence as 1, 2, 4, 7, 11, 16, 22, 29, 37, ... , with offset 0, while MathworldMathWorldMathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...
gives the sequence as 2, 4, 7, 11, 16, 22, 29, 37, ... (implied offset 1). It can be argued that making no cuts to the pancake is technically a number of cuts, namely n = 0. But it can also be argued that an uncut pancake is irrelevant to the problem. Although the offset is a required field, some contributors don't bother to check if the default offset of 0 is appropriate to the sequence they are sending in. The internal format actually shows two numbers for the offset. The first is the number described above, while the second represents the index of the first entry (counting from 1) that has an absolute value greater than 1. This second value is used to speed up the process of searching for a sequence. Thus , which starts 1, 1, 1, 2 with the first entry representing a(1) has 1, 4 as the internal value of the offset field.
Author(s)
- The author(s) of the sequence is (are) the person(s) who submitted the sequence, even if the sequence has been known since ancient times. The name of the submitter(s) is given first name (spelled out in full), middle initial(s) (if applicable) and last name; this in contrast to the way names are written in the reference fields. The e-mail address of the submitter is also given, with the @ character replaced by "(AT)" with some exceptions such as for associate editors or if an e-mail address does not exist. For most sequences after A055000, the author field also includes the date the submitter sent in the sequence.
Extension
- Names of people who extended (added more terms to) the sequence, followed by date of extension.
Searching the OEIS
The previous version of the main look-up page of the OEIS offered three ways to look up sequences, and the right radio button had to be selected. There was an advanced look-up page, but its usefulness has been integrated into the main look-up page in a major redesign of the interface in January 2006.Enter a sequence
Enter a few terms of the sequence, separated by either spaces or commas (or both).You can enter negative signs, but they will be ignored. For example, 0, 3, 7, 13, 20, 28, 36, 43, 47, 45, 32, 0, −64, n2 minus the nth Fibonacci number, is a sequence that is technically not in the OEIS, but the very similar sequence 0, −3, −7, −13, −20, −28, −36, −43, −47, −45, −32, 0, 64, is in the OEIS and will come up when one searches for its reversed signs counterpart.
However, the search can be forced to match signs by using the prefix "signed:" in the search string. This is especially useful for sequences like that consist exclusively of positive and negative ones.
One can enter as little as a single integer or as much as four lines of terms. Sloane recommends entering six terms, a(2) to a(7), in order to get enough results, but not too many results. There are cases where entering just one integer gives precisely one result, such as 6610199 brings up just , the strobogrammatic prime
Strobogrammatic prime
A strobogrammatic prime is a prime number that, given a base and given a set of glyphs, appears the same whether viewed normally or upside down. In base 10, given a set of glyphs where 0, 1 and 8 are symmetrical around the horizontal axis, and 6 and 9 are the same as each other upside down, A...
s which are not palindromic). There are also cases where one can enter many terms and still not narrow the results down very much.
Enter a word
Enter a string of alphanumerical characters. Certain characters, like accented foreign letters, are not allowed. Thus, to search for sequences relating to Znám's problemZnám's problem
In number theory, Znám's problem asks which sets of k integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. Znám's problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972, although other...
, try enter it without the accents: "Znam's problem." The handling of apostrophes has been greatly improved in the 2006 redesign. The search strings "Pascal's triangle
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
," "Pascals triangle" and "Pascal triangle" all give the desired results.
To look up most polygonal number
Polygonal number
In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots were thought of as alphas . These are one type of 2-dimensional figurate numbers.- Definition and examples :...
s by word, try "n-gonal numbers" rather than "Greek prefix-gonal numbers" (e.g., "47-gonal numbers" instead of "heptaquartagonal numbers"). Beyond "dodecagonal numbers," word searching with the Greek prefixes might fail to yield the desired results.