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The On-Line Encyclopedia of Integer Sequences

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ID Number: A004001 (Formerly M0276 and N0101))
Sequence:  1,1,1,0,0,1,1,2,0,0,1,1,2,2,2,4,1,2,0,0,3,4,6,6,8,8,10,10,9,9,7,5,2,0,
Signed:    1,-1,-1,0,0,1,1,2,0,0,-1,-1,-2,-2,-2,-4,-1,-2,0,0,3,4,6,6,8,8,10,10,9,9,7,
Name:      Bell or exponential numbers: ways of placing n labeled balls into n
           indistinguishable boxes.
Comments:  On first day, each gossip has his own tidbit. On each successive day, disjoint pairs of
           gossips may share tidbits (over the phone). After a(n) days, all gossips have all
References R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley,
           Reading, MA, 1990, p. 329.
           C. L. Mallows, Conway's challenge sequence, Amer. Math. Monthly, 98 (1991), 5-20.
Links:     D. E. Iannucci and D. Mills-Taylor, On Generalizing the Connell Sequence,
           J. Integer Sequences, Vol. 2, 1999, #7.
Formula:   a(n)=(1/4)*n^2*(n^2+3).
Example:   a(24)=4 because we can form 2, 4, 24 and 42.
Maple:     a:=proc(n) option remember; if n<=2 then 1 else a(a(n-1))+a(n-a(n-1)); fi; end;
Math'ca:   dtn[L_]:=Fold[2#1+#2&,0,L]; f[n_]:=dtn[Reverse[1-IntegerDigits[n,2]]];
Program:   (PARI.2.0.11) direuler(p=2,101,1/(1-(kronecker(5,p)*(X-X^2))-X))
See also:  Cf. A039800
Keywords:  sign,nice,easy
Offset:    5
Author(s): Christian G. Bower (bowerc(AT)usa.net), February 8, 1999.
Extension: Extended by David Wilson (wilson(AT)ctron.com), Mar 10, 1999.

Explanation of the Different Lines

ID Number

  • The A-number (for example A000108) is the absolute catalogue number of the sequence. It consists of A followed by 6 digits.
  • Some sequences also have a 4-digit M-number, such as M1459, which is the number they carried in "The Encyclopedia of Integer Sequences" by N.J.A. Sloane and S. Plouffe, Academic Press, San Diego, CA, 1995.
  • Some older sequences also have a 4-digit N-number, such as N0577, which is the number they carried in the "Handbook of Integer Sequences", by N. J. A. Sloane, Academic Press, NY, 1973.


  • These lines give the beginning of the sequence.
    • For example:     0,1,1,2,3,5,8,13,21,34,55,89,144,...
  • Ideally the entry gives enough terms to fill about three lines on the screen.
  • If the sequence contains negative numbers then the Sequence lines give the absolute values of the terms.
  • The terms must be integers.
  • If the terms are fractions, then the numerators and denominators appear as separate sequences, labeled with the Keyword "frac", and with links connecting the two sequences.
  • Only sequences that are well-defined and of general interest are included.


  • These lines give the beginning of the sequence, in the case that some of the terms are negative.
  • The Ramanujan numbers, sequence A000594, are a famous example.
  • In such cases the Sequence lines give the beginning of the sequence as unsigned numbers. These numbers match one-to-one with the numbers in the "Signed" lines.


  • The "Name" line gives a brief description or definition of the sequence.
    • For example:     The even numbers.
  • In the description, a(n) usually denotes the n-th term of the sequence, and n is a typical subscript.
    • For example:     a(n) = a(n-1) + a(n-3).
  • In some cases however n denotes a typical term in the sequence.
    • For example:     n and n+1 have the same number of divisors.


  • Additional remarks about the sequence that do not fit into any of the other lines (additional situations where the sequence occurs, for instance).


  • References where information about the sequence can be found.
  • Whenever possible the reference gives full bibliographical information:
    • For an article in a journal: author(s), title of article, name of journal, volume, issue number if relevant, year, starting and ending page numbers, etc.
    • For a book: author(s), title, publisher, place, year, edition, page numbers where sequence appears, etc.
    • For an article in a book: author(s), title of article, page numbers, editors' names, title of book, publisher, place, year, etc.


  • Links related to this sequence
  • Preferred format:

    J. B. Smith, < a href = " http : // www.this.that.com/etc/etc.html ">Title< /a >

  • - spaces have been inserted to make it visible, but you should not insert any spaces of course.

    In other words, the format is

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

  • Web page addresses can change very quickly, so if you find a link that is broken, please add a comment to that link — you might say, for example, [broken link? - ~~~~] (the four tildes will be transformed into your signature).


  • These lines give formulae, recurrences, generating functions, etc. for the sequence.
  • a(n) usually denotes the n-th term of the sequence, and n is a typical subscript.
  • Note that the Offset line gives the value of n corresponding to the first term shown.
    • An example of an explicit formula:     a(n) = n^2 + n + 1.
    • An example of a recurrence:     a(n+1) = 2 * a(n) - (-1)^n * 3.
  • The ordinary generating function (G.f.) for a sequence a(0), a(1), a(2), ... is the formal power series
            A(x) = a(0) + a(1)*x + a(2)*x^2 + a(3)*x^3 + ...
    • An example of an ordinary generating function:     G.f.: A(x) = 1/(1-x)^4.
    • Usually one can think of an ordinary generating function as a Taylor series, and extract the nth coefficient by differentiating A(x) n times, setting x = 0, and dividing by n!. Computer algebra languages such as Maple make this easy - one simply says (for example) series(A,x,100).
  • The exponential generating function (E.g.f.) for a sequence a(0), a(1), a(2), ... is the formal power series
                    a(0)   a(1)*x   a(2)*x^2   a(3)*x^3   a(4)*x^4
            A(x) =  ---- + ------ + -------- + -------- + -------- + ...
                     1       1         2          6          24
    where the numbers in the denominators are the factorial numbers n! = 1*2*3*4*...*n, Sequence A000142.
    • An example of an exponential generating function:     E.g.f.: A(x) = exp(exp(x)-1).


  • These lines give expanded information or examples to illustrate the initial terms of the sequence.
    • For instance:     4=2^2, so a(4)=1;   5=1^2+2^2=2^2+1^2, so a(5)=2.
  • If the sequence is formed from the coefficients of a power series, this line can be used to show the beginning of the series.
    • For instance:     1+3600*q^3+101250*q^4+...
  • If the sequence is formed from the decimal expansion or continued fraction expansion of a real number, this line may show the actual decimal expansion.
    • For instance:     3.141592653589793238462643383279502884...
  • If the sequence is formed by reading the rows of an array, this line may show the beginning of the array (see the Keywords "tabl" and "tabf" below.)
    • For instance:     {1}; {1,1}; {1,2,1}; {1,3,3,1}; {1,4,6,4,1}; ...


  • These lines give Maple code to produce the sequence. Examples:
    • f:=i->if isprime(i) then 1 else 0; fi; [seq(f(i),i=0..100)];
    • for i from 1 to 100 do if isprime(i) then print(nops(factorset(i-1))); fi; od;


  • These lines give Mathematica code to produce the sequence. For example:
    • Table[ If[ n==1,1,LCM@@Map[ (#1[ [ 1 ] ]-1)*#1[ [ 1 ] ]^(#1[ [ 2 ] ]-1)&, FactorInteger[ n ] ] ],{n,1,70} ]


  • These lines give a program in some other language that will produce the sequence. Examples:
    • (PARI) v=[];for(n=0,60,if(isprime(n^2+n+41),v=concat(v,n),));v
    • (MAGMA) R := ReedMullerCode(2,7); print(WeightEnumerator(R));
    • (Sage) CuspForms( Gamma1(1), 12, prec=100).0;

See also

  • These lines gives cross-references to related sequences. Examples:
    • Cf. A006546, A007104, A007203.
    • a(n) = A025582(n)^2+1.
  • Sequence in context. This line show the three sequences immediately before and after the sequence in the lexicographic listing. Example:
    • Sequence in context: A036656 A000055 A006787 this_sequence A036648 A047750 A072187

  • Adjacent sequences. This line show the three sequences whose A-numbers are immediately before and after the A-number of the sequence. Example:
    • Adjacent sequences: A000989 A000990 A000991 this_sequence A000993 A000994 A000995


These lines give keywords describing the sequence. At present the following keywords are in use.
  • base: Sequence is dependent on base used
  • bref: Sequence is too short to do any analysis with
  • changed: A sequence that was changed in the last two or three weeks (this keyword is set automatically)
  • cofr: A continued fraction expansion of a number
  • cons: A decimal expansion of a number
  • core: An important sequence
  • dead: An erroneous or duplicated sequence (the table contains a number of incorrect sequences that have appeared in the literature, with pointers to the correct versions)
  • dumb: An unimportant sequence
  • dupe: Duplicate of another sequence
  • easy: It is easy to produce terms of this 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 rational numbers
  • full: The full sequence is given (implies that the sequence is finite)
  • hard: Next term is not known and may be hard to find. Would someone please extend this sequence?
  • hear: A sequence worth listening to.
  • less: This is a less interesting sequence and is less likely to be the one you were looking for.
  • look: A sequence with an interesting graph.
  • more: More terms are needed! Would someone please extend this sequence? We need enough terms to fill about three lines on the screen.
  • mult: Multiplicative: a(mn)=a(m)a(n) if g.c.d.(m,n)=1
  • new: New (added or modified within last two weeks, roughly; this keyword is set automatically)
  • nice: An exceptionally nice sequence
  • nonn: A sequence of nonnegative numbers (more precisely, all the displayed terms are nonnegative; it is not excluded that later terms in the sequence become negative)
  • obsc: Obscure, better description needed
  • probation: Included on a provisional basis, but may be deleted later at the discretion of the editor.
  • 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: A regular array of numbers, such as Pascal's triangle, made into a sequence by reading it row by row
  • uned: Not edited. The editors normally check all incoming sequences to make sure 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 needs editing. If you can help by editing the entry, please do so!
  • unkn: Little is known; an unsolved problem; anyone who can find a formula or recurrence is urged to add it to the entry.
  • walk: Counts walks (or self-avoiding paths)
  • word: Depends on words for the sequence in some language
  • For further information about the keywords, see here.


  • This line usually gives the subscript of the first term in the sequence.
    • For example: the Fibonacci numbers F(0), F(1), F(2), ... begin     0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,... and the subscript of the initial term is 0,
      so the "Offset" line is     0
  • If the sequence gives the decimal expansion of a constant, the offset is the number of digits before the decimal point.
    • For example, the speed of light is     299792458 (m/sec),     giving the sequence 2,9,9,7,9,2,4,5,8,     with offset 9    .
  • In the internal format, there is a second offset, which says which term (counting from the left, and starting with 1), first exceeds 1 in absolute value. This is set to 1 if all the terms are 0 or +-1.


  • This line gives the name of the person or persons who contributed the sequence. For example:     Clark Kimberling. This name will usually be an active link to the user page of the submitter on the OEIS wiki.


  • E stands for Extensions, Errors or Edited. These lines contain information about sequences that have been significantly extended, errors that have been corrected, or entries in the database that have been edited by someone.
  • The errors might be in an earlier version of the entry in the database or in the published literature.
  • Examples:
    • Corrected and extended by Henry Bottomley (se16(AT)btinternet.com), Jan 01 2002
    • The sixth term is incorrect in the book by Smith and Jones.
    • Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Jan 02 2002


  • The database also contains a number of sequences based on triangular or square arrays, such as Pascal's Triangle:

    1 1
    1 2 1
    1 3 3 1
    1 4 6 4 1
    1 5 10 10 5 1
    ... ... ... ... ... ... ...

    When read by rows this produces the sequence 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, ..., Sequence A007318.

  • Square arrays are usually read by anti-diagonals. For example, the Nim-addition table:

    when read by anti-diagonals produces the sequence 0, 1, 1, 2, 0, 2, 3, 3, 3, 3, 4, 2, 0, 2, 4, ..., Sequence A003987.

  • The typical term in these arrays is usually denoted by a(n,k) (sometimes T(n,k)) in the Formula lines.
  • The Example lines for these sequences usually show the beginning of the two-dimensional array.
  • These sequences are usually indicated by the Keyword tabl.
  • Some ordinary (one-dimensional) sequences also have the keyword tabl, indicating that they can also be regarded as arrays.
  • The Keyword tabf indicates a sequence formed by reading a "funny-shaped" array. More precisely, this a sequence where the array cannot be recovered simply by breaking up the sequence into chunks of successive lengths 1, 2, 3, 4, 5, ... Typically one has to use chunks of lengths 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ... or 1, 3, 5, 7, 9, 11, 13, ... for "tabf" sequences. See A028297 and A027113 for examples.

    Wolfdieter Lang has a very nice program that will format both "tabl" and "tabf" sequences. It is not finished yet but there is a preliminary version at http://www-itp.physik.uni-karlsruhe.de/~wl/Anumbertest.html

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Last modified October 23 06:43 EDT 2014. Contains 248415 sequences.