Internal Format Used In
This file describes the internal format
used in
The OnLine 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 6digit
identification number (or catalogue number) of the sequence.
Each sequence gets a unique Anumber.
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. 14441578.
%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. 100120, 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:
Use "Back" to return to this page.
 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.
 An example with many comments, references and links:
A000108.
 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,
crossreferenced 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,
crossreferenced 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 crossreferenced
to each other in %Y lines. For example:
A000367
and
A002445.
For even more examples,
enter an arbitrary Anumber (e.g. A005132) here and click "Submit":
Use "Back" to return to this page.
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 sending new sequences, start numbering
them A000001, A000002, A000003, etc.
Use 6digit numbers. I will renumber them.
If you are planning to send many sequences, I can issue you a block of Anumbers
and you can number them yourself.
 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.
(If the sequence contains negative numbers then these
give the absolute values of the terms.)
 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 welldefined 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
%V, %W and %X lines
 The %V, %W and %X lines are present only if the sequence contains
negative numbers, and if so they give the signed terms of the sequence.
 Example: the Ramanujan numbers
(A000594)
is a famous sequence with signs. The %S, %T, %U lines
give the beginning of the sequence as unsigned numbers,
then the %V, %W, %X lines
give the signed version:
%S A000594 1,24,252,1472,4830,6048,16744,84480,113643,115920,534612,
%T A000594 370944,577738,401856,1217160,987136,6905934,2727432,10661420,
%U A000594 7109760,4219488,12830688,18643272,21288960,25499225,13865712
%V A000594 1,24,252,1472,4830,6048,16744,84480,113643,115920,534612,
%W A000594 370944,577738,401856,1217160,987136,6905934,2727432,10661420,
%X A000594 7109760,4219488,12830688,18643272,21288960,25499225,13865712
 The %V line should match the %S line, the %W line should
match the %T line, and the %X line should match the %U line.
%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 nth 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, LowTemperature Series Expansions
for the Spin1 Ising Model, J. Phys. A. 27 (1994) 69877006.
%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 selfcompositions 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 lessthan 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 nth 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/(12*x^2x^3).
%F A014551 a(n+1) = 2 * a(n)  (1)^n * 3.
%N A030033 a(n+1)= Sum a(k)a(nk), k = 0 ... [2n/3].
 Note that the %O line (see below)
gives the initial value of n.
%Y = Crossreferences to other sequences
%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 Anumbers are
immediately before and after the Anumber of the sequence.
Example:
%Y A000112 Adjacent sequences: A000109 A000110 A000111 this_sequence A000113
A000114 A000115
%A = Author, submitter or other Authority:
Required!
 Give your name and email address.
 Example:
%A A023600 Clark Kimberling (ck6(AT)evansville.edu)
 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(i1))); fi; od;
%t A011773 Table[If[n==1,1,LCM@@Map[ (#1[1]]1)*#1[1]]^(#1[2]]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
 dead: an erroneous 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
The %V, %W, %X lines (matching the %S,T,U lines)
give the signed sequence
 tabf: An irregular (or funnyshaped)
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: crossreferences 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 selfavoiding paths)
 word: depends on words for the sequence in some language
 Examples:
%K A029403 nonn
%K A002654 core,easy,nonn
%K A024022 sign
%C = Comments
 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 2dim. 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 kdigit number is one of the k! permutations of the
digits that does not begin with 0.
%I A000001 Identification line (required)
%S A000001 First line of unsigned sequence (required)
%T A000001 2nd line of unsigned sequence.
%U A000001 3rd line of unsigned sequence.
%V A000001 First line of signed sequence.
%W A000001 2nd line of signed sequence.
%X A000001 3rd line of signed 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 Crossreferences 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)
%C A000001 Comments.
