On-Line Encyclopedia of Integer Sequences

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The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web.

OEIS records information on integer sequences of interest to both professional mathematicians and amateurs, and is widely cited. It contains over 100,000 sequences as of December 2005, making it the largest database of its kind.

Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more. The database is searchable by keyword and by subsequence.


Neil Sloane started collecting integer sequences as a student in the mid-1960's 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 012648550X), containing 2,400 sequences.
  2. The Encyclopedia of Integer Sequences (1995, ISBN 0125586302), containing 5,487 sequences.

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 (1995), and soon after as a web service (1996). 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 2002 a board of associate editors and volunteers has helped maintain the database.

As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998.

In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, Template:OEIS2C. In 2006, the user interface was overhauled and more advanced search capabilities were added.


Besides integer sequences strictly speaking, OEIS also catalogued 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, , is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 (Template:OEIS2C) and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5 (Template:OEIS2C).

Important irrational numbers such as π = 3.1415926535897 ... are catalogued under their decimal digit sequence: 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7 ... (Template:OEIS2C).


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 zeroes, 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 sequence a.

Special meaning of zero

Zero is often used to represent non-existent sequence elements. For example, Template:OEIS2C 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 (Template:OEIS2C) 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.

Lexicographic ordering

The OEIS maintains the lexicographic order of the sequences, so each sequence has a predecessor and a successor (its "context"). OEIS normalizes the sequences for lexicographic ordering, (usually) ignoring initial zeroes or ones and also the sign of each element. Sequences of weight distributions of codes tend to omit periodically recurring zeroes.

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 . 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.


Very early in the history of the OEIS, many people suggested sequences derived from the placement of sequences in the OEIS itself. "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 Template:OEIS2C (later Template:OEIS2C) "a(n) = n-th term of sequence A_n." This sequence spurred progress on finding more terms of Template:OEIS2C. For larger n that correspond to sequences that are finite and given in full (keywords "fini" and "full"), term a(n) of A091967 is undefined.

Template:OEIS2C 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 "Is n in sequence An?" and the delightfully paradoxical sequences Template:OEIS2C, n is in An, and Template:OEIS2C, n is not in An. Thus, the composite number 2808 is in A053873 because Template:OEIS2C is the sequence of composite numbers, while the non-prime 40 is in A053169 because it's not in A000040, the prime numbers. The paradox is, which sequences do 53169 and 53873 belong to?

An abridged example of a typical OEIS entry

This entry, Template:OEIS2C, was chosen because, with the exception of a Maple program, it contains every field an OEIS entry can have.

ID Number: A046970
URL:       http://www.research.att.com/projects/OEIS?Anum=A046970
Sequence:  1,3,8,3,24,24,48,3,8,72,120,24,168,144,192,3,288,24,360,72,
Signed:    1,-3,-8,-3,-24,24,-48,-3,-8,72,-120,24,-168,144,192,-3,-288,
Name:      Generated from Riemann Zeta function: coefficients in series
              expansion of Zeta(n+2)/Zeta(n).
Comments:  ...
           Apart from signs also Sum_{d|n} core(d)^2*mu(n/d) where core(x) is
              the squarefree part of x. - Benoit Cloitre
              (abcloitre(AT)modulonet.fr), May 31 2002
References M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,
              Dover Publications, 1965, pp. 805-811.
Links:     Wikipedia, Riemann zeta function.
Formula:   Multiplicative with a(p^e) = 1-p^2. a(n) = Sum_{d|n} mu(d)*d^2.
Example:   a(3) = -8 because the divisors of 3 are {1, 3}, and mu(1)*1^2 + mu(3)*3^2 =
           a(4) = -3 because the divisors of 4 are {1, 2, 4}, and mu(1)*1^2 +
              mu(2)*2^2 + mu(4)*4^2 = -3
Math'ca:   muDD[d_] := MoebiusMu[d]*d^2; Table[Plus @@ muDD[Divisors[n]], {n,
              60}] (Lopez)
Program:   (PARI) A046970(n)=sumdiv(n,d,d^2*moebius(d)) (Benoit Cloitre)
See also:  Sequence in context: A016623 A046543 A035292 this_sequence A058936
              A002017 A086179
           Adjacent sequences: A046967 A046968 A046969 this_sequence A046971
              A046972 A046973
           Cf. A027641 and A027642.
Keywords:  sign,mult
Offset:    1
Author(s): Douglas Stoll, dougstoll(AT)email.msn.com
Extension: Corrected and extended by Vladeta Jovovic (vladeta(AT)Eunet.yu),
              Jul 25 2001

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). 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.

A059097 Numbers n such that the binomial coefficient C(2n,n) is not divisible by the square of an odd prime. January 1, 2001
A060001 Fibonacci(n)!. March 14, 2001
A066288 Number of 3-dimensional polyominoes (or polycubes) with n cells and symmetry group of order exactly 24. January 1, 2002
A075000 Smallest number such that n*a(n) is a concatenation of n consecutive integers ... August 31, 2002
A078470 Continued fraction for Zeta(3/2) January 1, 2003
A080000 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i February 10, 2003
A090000 Length of longest contiguous block of 1's in binary expansion of n-th prime. November 20, 2003
A091345 Exponential convolution of A069321(n) with itself, where we set A069321(0)=0. January 1, 2004
A100000 Marks from the 22000-year-old Ishango bone from the Congo. November 7, 2004
A102231 Column 1 of triangle A102230, and equals the convolution of A032349 with A032349 shift right. January 1, 2005
A110030 Number of consecutive integers starting with n needed to sum to a Niven number. July 8, 2005
A112886 Triangle-free positive integers. January 12, 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.


The URL field gives the preferred format for the URL to link to the sequence in question, to simplify cut and paste.


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.

Any negative signs are stripped from this field, and the values with signs are put in the Signed field.


The signed field is almost the same thing as the sequence field except that it shows negative signs. This field is only included for sequences that have negative values. Any entry with this field must have the keyword "sign".


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, (Template:OEIS2C) 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, Lekray 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 (Template:OEIS2C) and second Bessel polynomials (Template:OEIS2C) in a comment to A003215.

If no name is given for a comment, the comment was made by the original submitter of the sequence.

Maple, Mathematica and other programs

Maple and Mathematica are the preferred programs for calculating sequences in the OEIS, and they both get their own field labels, "Maple" and "Mathematica." Any other program gets a generic "Program" field label and the name of the program in parentheses. The OEIS has programs in PARI, Magma, Matlab and even Microsoft Excel.

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:

A016623 3, 8, 3, 9, 4, 5, 2, 3, 1, 2, ... Decimal expansion of ln(93/2).
A046543 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).
A035292 1, 3, 8, 3, 12, 24, 16, 3, 41, 36, 24, ... Number of similar sublattices of Z^4 of index n^2.
A046970 1, -3, -8, -3, -24, 24, -48, -3, -8, 72, ... Generated from Riemann Zeta function...
A058936 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.
A002017 1, 1, 1, 0, -3, -8, -3, 56, 217, 64, -2951, -12672, ... Expansion of exp(sin x).
A086179 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.


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 ... Template:OEIS2C are prime numbers regardless of base, but they are palindromic specifically in base 10, most of them are not palindromic in binary. It's interesting to note, however, that some sequences rate this keyword depending on how they're defined. For example, 3, 7, 31, 127, 8191, 131071, ... Template:OEIS2C 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, Template:OEIS2C, 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 is defined as being for erroneous sequences that have appeared in papers or books. But a random sampling of sequences with this keyword turns up a lot of sequences that are duplicates of other sequences and the name field reads "Same as Am," "Essentially the same as Am" or "Duplicate of Am." For example, Template:OEIS2C, which is the same as A000668.
  • dumb One of the more subjective keywords, for "unimportant sequences," which may or may not directly relate to mathematics. Template:OEIS2C, "Mix digits of pi and e." is one example of the former, and Template:OEIS2C, "Numbers on a computer keyboard, read in a spiral." is an example of the latter.
  • dupe A keyword for duplicate sequences, but in practice, the keyword "dead" is often used instead.
  • eigen A sequence of eigenvalues.
  • 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, ... Template:OEIS2C, 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).
  • fini The sequence is finite, although it might still contain more terms than can be displayed. For example, the sequence field of Template:OEIS2C 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 Template:OEIS2C, where the sequence of numerators would be Template:OEIS2C. 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 Template:OEIS2C, 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 spheres can touch another sphere of the same size?" Template:OEIS2C lists the first ten known solutions.
  • more The next few terms of the sequence are not known. This keyword usually goes hand in hand with the keyword "hard."
  • 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. One sequence with this keyword, Template:OEIS2C has been looked at by at least one other OEIS contributor who was unable to reproduce the results given by the original submitter.
  • 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, Template:OEIS2C, "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, Template:OEIS2C.
  • 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, Template:OEIS2C, which was presented to an 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 ... Template:OEIS2C, "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, nonn and sign, and possibly core and new (unless a new brach of mathematics comes up with its own set of sequences). But dumb and nice are not necessarily mutually exclusive, cf. Template:OEIS2C.


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 Template:OEIS2C, 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 Template:OEIS2C, "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, ... Template:OEIS2C, 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 Template:OEIS2C, which starts 1, 1, 1, 2 with the first entry representing a(1) has 1, 4 as the internal value of the offset field.


The author of the sequence is the person who submitted the sequence, even if the sequence has been known since ancient times. The name of the submitter 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 of the submitter is also given, with the @ character replaced by "(AT)". For most sequences after A055000, the author field also includes the date the submitter sent in the sequence. But when the submitter is Neil Sloane himself, the author field just says "njas," Sloane's initials.

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 "sign:" in the search string. This is especially useful for sequences like Template:OEIS2C 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 1990661 brings up just Template:OEIS2C, the strobogrammatic primes). And there are also cases where one can enter as many as 30 terms and still not narrow the results down all that much. For example, if you enter the first 20 integers, A000027 will not come up or even be mentioned. Entering the first 40 integers finally brings up A000027 as the first result.

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 entering "Znam" rather than "Znam's problem." However, 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 Template:OEIS2C, 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 Template:OEIS2C, concatenation of numbers from n down to 1.

Errors or problems in the OEIS

For a database of its magnitude, the OEIS is relatively free of errors. But being the work of humans, it is inevitable that it will have problems and even mistakes.

Computational and typographical errors in the sequence field are extremely rare. Many sequence entries include computer programs, and many people devote considerable time to double-checking and extending results in the OEIS. It is possible that some sequences have large probable primes (or numbers corresponding to large probable primes) that will turn out to be pseudoprimes rather than actual primes, but these cases are almost always noted in a comment field. The more common mistakes in the OEIS occur in fields other than the sequence or signed field.

  • Missing keywords. Many multiplicative sequences are missing the keyword "mult." (Any sequence with the keyword "full" but not "fini" would constitute an obvious example of a missing keyword, but this kind of mistake appears to be extremely rare in the OEIS).
  • Wrong offset. When submitting a sequence using the form provided by Sloane, the default offset is 0. Many contributors don't change that offset, not bothering to consider whether that offset might be correct for the sequence they are sending in.
  • Duplicates. Some sequences are duplicated (the sequence fields are exactly the same, the name fields are worded slightly different, but the formula is exactly the same, and there is no cross-reference between the two sequences). This sometimes causes connections between different properties of a sequence to be overlooked.

External links

Papers on OEIS by Neil Sloane

Other references