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OEIS records information on integer sequences of interest to both professional Mathematician s 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 Term s of the sequence, Keyword s, mathematical motivations, literature links, and more. The database is Searchable by keyword and by subsequence. HISTORY 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: #''A Handbook of Integer Sequences'' (1973, ISBN 012648550X), containing 2,400 sequences. #''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, . In 2006, the user interface was overhauled and more advanced search capabilities were added. NON-INTEGERS Besides integer sequences strictly speaking, OEIS also catalogued sequences of Fraction s, the digits of Transcendental Number s, Complex Number s 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 () and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5 (). 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 ... (). 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 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 ''n''th term of sequence ''a''. Special meaning of zero Zero is often used to represent non-existent sequence elements. For example, enumerates the "smallest prime of ''n''&2 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 () 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 Number s, the Palindromic Prime s, 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. SELF-REFERENTIALITY 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 (later ) "a(n) = n-th term of sequence A_n." This sequence spurred progress on finding more terms of . 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. lists the first term given in sequence A''n'', but it needs to be updated from time to time because of changing opinions on offsets. Listing instead term ''a''(1) of sequence A''n'' 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 A''n''?" and the delightfully paradoxical sequences , ''n'' is in A''n'', and , ''n'' is ''not'' in A''n''. 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. The paradox is, which sequences do 53169 and 53873 belong to? 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.
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