# On-Line Encyclopedia of Integer Sequences

﻿
On-Line Encyclopedia of Integer Sequences
URL oeis.org Neil Sloane 1996

The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane's, is an online database of integer sequences, 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.[1] OEIS records information on integer sequences of interest to both professional mathematicians and amateurs, and is widely cited. As of 5 January 2011 it contains over 181,050 sequences, making it the largest database of its kind. Each entry contains the leading terms of the sequence, keywords, 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.

## History

Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics. The database was at first stored on punch cards. He published selections from the database in book form twice:

1. 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.
2. The Encyclopedia of Integer Sequences with Simon Plouffe (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.[2] 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.[3] 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.[4]

## Non-integers

Besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers 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, A006842) 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, ... ()).

## Conventions

The OEIS is currently limited to plain ASCII 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.

### 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 square 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 Nϕ(m) () 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 order 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 numbers, the palindromic primes, the Fibonacci sequence, the lazy caterer's sequence, and the coefficients in the series expansion of $\textstyle {{\zeta(n + 2)} \over {\zeta(n)}}$. 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.[5] 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 paradox.

## An abridged example of a typical OEIS entry

This entry, , was chosen because, with the exception of a Maple 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*(x[[i]][[1]]^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


## Entry fields

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.
 Numbers n such that the binomial coefficient C(2n, n) is not divisible by the square of an odd prime. January 1, 2001 Fibonacci(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 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."
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 () and second Bessel polynomials () in a comment to A003215.
References
References to printed documents (books, papers, ...)
Links, i.e. URLs, to online resources. These may be:
2. 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
3. links to images in the local database directories which often provide combinatorial background related to graph theory
4. references to applicable articles in journals
5. 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
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.
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:[6]
• 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 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, ... 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, , 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, 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 primes , 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 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 , 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, , "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, .
• 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 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 ... , "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 , 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 , "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, ... , 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 , 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 primes 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 problem, 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 numbers 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.

### Enter a sequence number

Enter the modern OEIS A number of the sequence, with the letter A and with or without zero-padding. As of 2006, the old M and N sequence numbers will yield the proper result as search strings, e.g., a search for M0422 will correctly bring up , the number of entries in nth row of Pascal's triangle not divisible by 3 (M0422 in the book The Encyclopedia of Integer Sequences) and not , concatenation of numbers from n down to 1.

## Notes

1. ^ Transfer of IP in OEIS to The OEIS Foundation Inc.
2. ^ Journal of Integer Sequences (ISSN 1530-7638)
3. ^ "Editorial Board". On-Line Encyclopedia of Integer Sequences.
4. ^ Neil Sloane (2010-11-17). "New version of OEIS".
5. ^ Sloane, N. J. A.. "My favorite integer sequences" (PDF). p. 10.
6. ^ "Explanation of Terms Used in Reply From". On-Line Encyclopedia of Integer Sequences.

## References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• On-Line Encyclopedia of Integer Sequences — Die On Line Encyclopedia of Integer Sequences (OEIS; deutsch Online Enzyklopädie der Zahlenfolgen) ist eine englischsprachige Datenbank von Folgen ganzer Zahlen (integer sequences), die über das Internet durchsucht werden kann. Sie ist ein oft… …   Deutsch Wikipedia

• On-Line Encyclopedia of Integer Sequences — Encyclopédie en ligne des suites de nombres entiers L encyclopédie en ligne des suites de nombres entiers (originellement en anglais On Line Encyclopedia of Integer Sequences, couramment abrégé sous le sigle OEIS) est un site web permettant d… …   Wikipédia en Français

• The On-Line Encyclopedia of Integer Sequences — Die On Line Encyclopedia of Integer Sequences (OEIS; deutsch Online Enzyklopädie der Zahlenfolgen) ist eine englischsprachige Datenbank von Folgen ganzer Zahlen (integer sequences), die über das Internet durchsucht werden kann. Sie ist ein oft… …   Deutsch Wikipedia

• Encyclopedia of Integer Sequences — Encyclopédie en ligne des suites de nombres entiers L encyclopédie en ligne des suites de nombres entiers (originellement en anglais On Line Encyclopedia of Integer Sequences, couramment abrégé sous le sigle OEIS) est un site web permettant d… …   Wikipédia en Français

• Integer sequence — In mathematics, an integer sequence is a sequence (i.e., an ordered list) 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. For example,… …   Wikipedia

• Integer sequence prime — In mathematics, an integer sequence prime is a prime number found as a member of an integer sequence. For example, the 8th Delannoy number, 265729, is prime. A challenge in empirical mathematics is to identify large prime values in rapidly… …   Wikipedia

• Integer — This article is about the mathematical concept. For integers in computer science, see Integer (computer science). Symbol often used to denote the set of integers The integers (from the Latin integer, literally untouched , hence whole : the word… …   Wikipedia

• Transposable integer — A summary of this article appears in Repeating decimal. The digits of some specific integers permute or shift cyclically when they are multiplied by a number n. Examples are: 142857 × 3 = 428571 (shifts cyclically one place left) 142857 × 5 =… …   Wikipedia

• Besondere Zahlen — sind zum einen Zahlen, die im Sinne der Zahlentheorie eine oder mehrere auffällige Eigenschaften besitzen. Außerdem haben viele Zahlen eine besondere Bedeutung in der Mathematik und/oder in Bezug auf die reale Welt. Diese letzteren Zahlen werden… …   Deutsch Wikipedia

• Q-Folge — In der Mathematik sind die Hofstadter Folgen Angehörige einer Familie ganzzahliger Folgen, die durch nichtlineare Differenzengleichungen beschrieben werden. Inhaltsverzeichnis 1 Hofstadter Folgen aus dem Buch Gödel, Escher, Bach: ein Endloses… …   Deutsch Wikipedia