On-Line Encyclopedia of Integer Sequences - Entry Fields

Entry Fields

See Format of OEIS Pages.

ID number

Every sequence in the OEIS has a serial number, 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.

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, ( A000578) 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 cevians within a triangle so that two of its sides are each n-partitioned," while Sloane points out the unexpected relationship between centered hexagonal numbers ( A003215) and second Bessel polynomials ( A001498) in a comment to A003215.

References

References to printed documents (books, papers, ...).

Links

Links, i.e. URLs, to online resources. These may be:

1. references to applicable articles in journals
2. links to the index
3. links to text files which hold the sequence terms (in a two column format) over a wider range of indices than held by the main database lines
4. links to images in the local database directories which often provide combinatorial background related to graph theory
5. others related to computer codes, more extensive tabulations in specific research areas provided by individuals or research groups

Formula

Formulae, recurrences, generating functions, etc. for the sequence.

Example

Some examples of sequence member values.

Maple

Maple code.

Mathematica

Mathematica code.

Program

Maple and Mathematica 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 PARI 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:

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 ...  A002385 are prime numbers regardless of base, but they are palindromic 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 primes 3, 7, 31, 127, 8191, 131071, ...  A000668 does not rate "base" if defined as "primes of the form 2^n - 1." However, defined as "repunit primes in binary," the sequence would rate the keyword "base."
• bref "sequence is too short to do any analysis with", for example,  A079243, 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 fraction.
• 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,  A088552 is the same as A000668.
• dumb One of the more subjective keywords, for "unimportant sequences," which may or may not directly relate to mathematics.  A001355, "Mix digits of pi and e." is one example of the former, and  A082390, "Numbers on a computer numpad, 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, ...  A000027, 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  A105417 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  A069257, where the sequence of numerators would be  A000012. 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 primes  A002267, 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?"  A001116 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 function. 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  A046970, 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 value function.
• tabf "An irregular (or funny-shaped) array of numbers made into a sequence by reading it row by row." For example,  A071031, "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 triangle read by rows,  A007318.
• 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,  A072036, which was presented to the Internet Oracle 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 ...  A005589, "Number of letters in the English name of n, excluding spaces and hyphens."

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  A073502, the magic constant for n×n magic square with prime entries (regarding 1 as a prime) with smallest row sums, is an example of a sequence with offset 3, and  A072171, "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 sequence, 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, ...  A000124, with offset 0, while Mathworld 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  A000001, 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.