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# Internal Format Used In

## The On-Line Encyclopedia of Integer Sequences

This file describes the internal format used in The On-Line Encyclopedia of Integer Sequences

[For a description of the standard (or beautified) format used in the web pages, click here.]

Each sequence is described by about 10 lines, each line beginning

```%x Aabcdef
```
where x is a letter (I, S, T, N, etc.) and abcdef is the 6-digit identification number (or catalogue number) of the sequence. Each sequence gets a unique A-number.

Here are two artificial examples, to illustrate the format used in the table (the abbreviations are explained below):

## A simple example:

```%I A007299
%S A007299 1,1,1,5,3,60,487
%N A007299 Hadamard matrices of order 4n.
%D A007299 M. Jones, The Catalan numbers, Amer. Math. Monthly, Vol. 256 (1939), pp. 1444-1578.
%K A007299 nonn,easy,more
%F A007299 a(n) = n^4 + 3*n.
%O A007299 1,4
%A A007299 Jane Smith (jsmith(AT)math.www.edu)
```

## A more complicated example:

```%I A000112 M1495 N0588
%S A000112 1,1,2,5,16,63,318,2045,16999,183231,2567284,46749427,1104891746,
%T A000112 33823827452
%N A000112 Partially ordered sets ("posets") with n elements.
%C A000112 A comment explaining the definition would go here.
%D A000112 A. Jones, Title of paper, Amer. Math. Monthly, vol. 21, pp. 100-120, 1991.
%D A000112 A. Jones, Further results on Euler's problem, preprint, 2002.
%H A000112 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000112 David Rusin, Finite Topologies
%H A000112 E. W. Weisstein, Harshad Numbers
%O A000112 0,3
%F A000112 a(n) = n^4 + 3*n.
%K A000112 nonn,hard,core
%p A000112 (n^2+n+3)*(n+29);
%Y A000112 Cf. A000798 (labeled topologies).
%Y A000112 Sequence in context: A022494 A079566 A059685 this_sequence A003149 A027046 A000522
%Y A000112 Adjacent sequences: A000109 A000110 A000111 this_sequence A000113 A000114 A000115
%A A000112 N. J. A. Sloane
```

## Further Examples and Style Guide:

• A simple formula: A051925. Note that the running variable is normally referred to as n.
• A simple recurrence: A046699. Note that the nth term is usually referred to as a(n).
• Numbers with some particular property: A051915. Here the typical term is usually referred to as n.
• Primes of a certain form: A045637.
• A partition function: A000837. Note that the words "The number of" are normally omitted if clear from the context.
• A sequence with signs: A000594.
• A continued fraction: A002852. It is customary to give the decimal expansion as a separate sequence, cross-referenced in a %Y line, and to give the beginning of the decimal expansion in a %e line.
• Decimal expansion of an important constant (as a sequence of single digits): A001620. It is customary to also give the true decimal expansion in a %e line. If possible also give the continued fraction for the number as a separate sequence, cross-referenced in a %Y line, and to give the beginning of the decimal expansion in a %e line.
• Theta series of a lattice: A004009. Note that if every other term is zero, then it is customary to omit these zeros.
• Weight distribution of a code: A010463. Note that if every other term (or every 4th term, etc) is zero, then it is customary to omit these zeros.
• An example with a complicated Maple program: A000022.
• A triangle of numbers read by rows: A008277. Note that the typical entry in the array is usually denoted by T(n,k) or T(n,m).
• A square array of numbers read by antidiagonals: A003987. Note that the typical entry in the array is usually denoted by T(n,k) or T(n,m).
• A sequence of fractions: Normally produces a pair of sequences, one giving the numerators (possibly with signs), and the other the denominators. They have keyword "frac" and are cross-referenced to each other in %Y lines. For example: A000367 and A002445.

## For even more examples,

enter an arbitrary A-number (e.g. A005132) here and click "Submit":

## Abbreviations Used

There is a summary at the end of this file.

## %I = Identification line:       Required!

• Annnnnn = absolute catalogue number of sequence
• Example:
```%I A012345
```
• When you submit a new sequence, it will automatically be assigned an A-number. You can also reserve a block of A-numbers, which is helpful if you are planning to submit a group of related sequences, so that you can cross-reference them.
• Mnnnn = number (if any) in "The Encyclopedia of Integer Sequences" by N.J.A. Sloane and S. Plouffe, Academic Press, San Diego, CA, 1995.
• Nnnnn = number (if any) in "Handbook of Integer Sequences", by N. J. A. Sloane, Academic Press, NY, 1973.

## %S, %T and %U lines

• The %S, %T and %U lines give the beginning of the sequence.
• The %S line (at least) is required.
• If possible, give enough terms to fill 3 lines on the screen. The numbers should be separated by commas, with no spaces or tabs. Don't give more than 3 lines. Label the 3 lines %S, %T and %U.
• At least 4 terms are required.
• The terms must be integers.
• If the terms are fractions, enter the numerators and denominators as separate sequences, use the keyword "frac", and put links in the %Y lines to link the two sequences together.
• The sequence should be well-defined and of general interest.
• Example (the Catalan numbers):
```
%S A000108 1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,
%T A000108 2674440,9694845,35357670,129644790,477638700,1767263190,
%U A000108 6564120420,24466267020,91482563640,343059613650,1289904147324

```

## %N = Name of sequence:       Required!

• The %N line gives a brief description or definition of the sequence. It must fit on one line (but the line can be fairly long).
• In the description, a(n) usually denotes the n-th term of the sequence, and n is a typical subscript.
• Only one %N line may appear.
• Here are 3 separate examples:
```
%N A000108 Catalan numbers: a(n) = C(2n,n)/(n+1) = (2n)!/(n!(n+1)!).

%N A000594 Ramanujan's tau function (or tau numbers).

%N A010085 Weight distribution of Hamming code of length 15 and minimal distance 3.

```

## %D = Detailed references.

• Put each reference on a separate line.
• There may be several such lines.
• Here are 3 separate examples:
```
%D A010109 I. G. Enting, A, J. Guttmann and I. Jensen, Low-Temperature Series Expansions
for the Spin-1 Ising Model, J. Phys. A. 27 (1994) 6987-7006.

%D A000925 A. Das and A. C. Melissinos, Quantum Mechanics: A Modern Introduction, Gordon
and Breach, 1986, p. 47.

%D A022818 W. C. Yang (wcyang(AT)cco.caltech.edu), Derivatives of self-compositions of
functions, preprint, 2002.

```

## %H = Links related to this sequence

• Put each link on a separate line.
• There may be several such lines.
• The lines must have the following form:
```
%H A036432 S. Colton, <a href="JIS/index.html#P99.1.2">
Refactorable Numbers - A Machine Invention,</a> J. Integer Sequences, Vol. 2, 1999, #2.

%H A001371 F. Ruskey, <a href=http://www.theory.cs.uvic.ca/~cos/inf/neck/NecklaceInfo.html">Counting Necklaces</a>

%H A027414 N. J. A. Sloane, <a href="transforms.html">Transforms</a>

```
(Except that you should use "pointed brackets" where I had to use "ampersand less-than semicolon"; and you should put the information on one long line, whereas I broke the lines to make them fit better on the screen).

In other words, please use this format:

%H A012345 Author, <a href="http://www.etc.etc/file">Title</a>

## %F = Formula (if not included in %N line)

• a(n) usually denotes the n-th term of the sequence, and n is a typical subscript.
• The ordinary generating function (G.f.) or exponential generating function (E.g.f.) is usually denoted by A(x).
• There may be several such lines.
• Here are 3 separate examples:
```
%F A008346 G.f.: 1/(1-2*x^2-x^3).

%F A014551 a(n+1) = 2 * a(n) - (-1)^n * 3.

%N A030033 a(n+1)= Sum a(k)a(n-k), k = 0 ... [2n/3].

```
• Note that the %O line (see below) gives the initial value of n.

## %Y = Cross-references to other sequences

• Examples:
```
%Y A003485 Cf. A003484.

%Y A007295 Cf. A006546, A007104, A007203.

%Y A005282 a(n) = A025583(n)^2+1.

```
• Sequence in context. This line show the three sequences immediately before and after the sequence in the lexicographic listing. Example:
```
%Y A000112 Sequence in context: A022494 A079566 A059685 this_sequence A003149
A027046 A000522

```
• Adjacent sequences. This line show the three sequences whose A-numbers are immediately before and after the A-number of the sequence. Example:
```%Y A000112 Adjacent sequences: A000109 A000110 A000111 this_sequence A000113
A000114 A000115

```

## %A = Author, submitter or other Authority:       Required!

• Example:
```
%A A023600 Clark Kimberling (ck6(AT)evansville.edu)

```

## %O = Offset a, b :Required!

• a is subscript of first term
• b gives position of first entry greater than or equal to 2 in magnitude (or 1 if no such entry exists)
• Examples: The Fibonacci numbers F(0), F(1), F(2), ... begin
```%S A000045 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,
```
and the 4th term is the first that is greater than 1, so here a = 0 and b = 4, and the %O line is:
```
%O A000045 0,4

```
• On the other hand, in this sequence:
```%S A010051 0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0
%N A010051 Characteristic function of primes: 1 if n is prime else 0.
```
no term exceeds 1, so b takes its default value of 1. n starts at 0, so a = 0, and the %O line is:
```
%O A010051 0,1

```

## %p, %t, %o = Computer program to produce the sequence

• %p = Maple
• %t = Mathematica
• %o = other computer language
• There may be several such lines, and the lines may be long.
• Examples:
```
%p A010051 f:=i->if isprime(i) then 1 else 0; fi; [seq(f(i),i=0..100)];

%p A008334 for i from 1 to 100 do if isprime(i) then print(nops(factorset(i-1))); fi; od;

%t A011773 Table[If[n==1,1,LCM@@Map[ (#1]-1)*#1]^(#1]-1)&, FactorInteger[n]]],{n,1,70}]

%o A002837 (PARI) v=[];for(n=0,60,if(isprime(n^2+n+41),v=concat(v,n),));v

%o A006006 (MAGMA) R := ReedMullerCode(2,7); print(WeightEnumerator(R));

```

## %E = Extensions and Errors

• Notes about sequences that have been significantly extended, etc.
• Also significant errors in the book or in the source.
• Examples:
```
%E A007097 15th term corrected by loria.fr!Paul.Zimmermann (Paul Zimmermann).

%E A010334 There is a typo at the n=6 term in the printed version of the paper.

```

## %e = examples

• Expanded information or examples to illustrate the initial terms of the sequence.
• If the sequence is the coefficients of a power series, the %e line can be used to show the beginning of the series.
• If the sequence is formed by reading the rows of an array, the %e line can show the beginning of the array (see the keyword "tabl" below.)
• Examples:
```
%e A002654 4=2^2, so a(4)=1; 5=1^2+2^2=2^2+1^2, so a(5)=2.

%e A027824 1+3600*q^3+101250*q^4+...

%e A007318 {1}; {1,1}; {1,2,1}; {1,3,3,1}; {1,4,6,4,1}; ...

```

## %K = Keywords:       Required!

• At the very least, indicate if the terms are all nonnegative ("nonn") or if there are negative numbers ("sign").

• base: dependent on base used for sequence
• bref: sequence is too short to do any analysis with
• cofr: a continued fraction expansion of a number
• cons: a decimal expansion of a number
• core: an important sequence
• dumb: an unimportant sequence
• dupe: duplicate of another sequence
• easy: it is very easy to produce terms of sequence
• eigen: an eigensequence: a fixed sequence for some transformation - see the files transforms and transforms (2) for further information.
• fini: a finite sequence
• frac: numerators or denominators of sequence of rationals
• full: the full sequence is given
• hard: next term not known, may be hard to find. Would someone please extend this sequence?
• less: reluctantly accepted
• more: more terms are needed! would someone please extend this sequence?
• mult: Multiplicative: a(mn)=a(m)a(n) if g.c.d.(m,n)=1
• new: New (added within last two weeks, roughly)
• nice: an exceptionally nice sequence
• nonn: a sequence of nonnegative numbers
• obsc: obscure, better description needed
• sign: sequence contains negative numbers
• tabf: An irregular (or funny-shaped) array of numbers made into a sequence by reading it row by row
• tabl: typically a triangle of numbers, such as Pascal's triangle, made into a sequence by reading it row by row.
• uned: Not edited. I normally edit all incoming sequences to check that:
• the sequence is worth including
• the definition is sensible
• the sequence is not already in the database
• the English is correct
• the different parts of the entry all have the correct prefixes: cross-references are in %Y lines, formulae in %F lines, etc.
• any %H lines are correctly formatted (this is easy to get wrong)
• etc.
The keyword "uned" indicates that this sequence was not edited, usually because of time pressure. Perhaps someone could edit this sequence and email me the result.
• unkn: little is known; an unsolved problem; anyone who can find a formula or recurrence is urged to let me know.
• walk: counts walks (or self-avoiding paths)
• word: depends on words for the sequence in some language

• Examples:
```
%K A029403 nonn

%K A002654 core,easy,nonn

%K A024022 sign

```

• Use this if you have a comment which does not fit into any of the other categories. Often used to give a more precise definition of the sequence, or to explain an unfamiliar word.
• Examples:
```
%C A002324 The hexagonal lattice is the familiar 2-dim. lattice in which each
point has 6 neighbors. This is sometimes called the triangular lattice.

%C A039997 a(n) counts substrings of digits of n which denote primes.

%C A046810 An anagram of a k-digit number is one of the k! permutations of the
digits that does not begin with 0.

```

## SUMMARY: all the possible lines:

```%I A000001 Identification line (required)
%S A000001 First line of sequence (required)
%T A000001 2nd line of sequence.
%U A000001 3rd line of sequence.
%N A000001 Name (required)
%D A000001 Detailed reference line.
%D A000001 Detailed references (2).
%H A000001 Link to other site.
%H A000001 Link to other site (2).
%F A000001 Formula.
%F A000001 Formula (2).
%Y A000001 Cross-references to other sequences.
%A A000001 Author (required)
%O A000001 Offset (required)
%E A000001 Extensions, errors, etc.
%e A000001 examples to illustrate initial terms.
%p A000001 Maple program.
%t A000001 Mathematica program.
%o A000001 Program in another language.
%K A000001 Keywords (required)