Integer Sequence

A sequence whose terms are integers. The most complete printed references for such sequences are Sloane (1973) and its update, Sloane and Plouffe (1995). Neil Sloane maintains the sequences from both these works in a vastly expanded on-line encyclopedia known as the On-Line Encyclopedia of Integer Sequences (https://www.research.att.com/~njas/sequences/). In this listing, sequences are identified by a unique 6-digit A-number. Sequences appearing in Sloane and Plouffe (1995) are ordered lexicographically and identified with a 4-digit M-number, and those appearing in Sloane (1973) are identified with a 4-digit N-number. To look up sequences by e-mail, send a message to either mailto:sequences@research.att.com or mailto:superseeker@research.att.com containing lines of the form lookup 5 14 42 132 ... (note that spaces must be used instead of commas).

Integer sequences can be analyzed by a variety of techniques (Sloane and Plouffe 1995, p. 26), including the application of a data compression algorithm (Bell et al. 1990), computation of the discrete Fourier transform (Loxton 1989), or searching for a linear recurrence equation connecting the terms or a generating function producing them. There are also a large number of transformations which relate integer sequences to one another, including the Euler transform, exponential transform, Möbius transform, and others (Bower, Sloane).

Closed forms for the terms of some sequences can be found in the Wolfram Language using the command FindSequenceFunction[seq].

In the Season 2 episode "Backscatter" (2006) of the television crime drama NUMB3RS, math genius Charlie Eppes poses a problem of identifying an integer sequence to his students, one of whom uses Sloane's Online Encyclopedia of Integer Sequences to find it.

REFERENCES:

Aho, A. V. and Sloane, N. J. A. "Some Doubly Exponential Sequences." Fib. Quart. 11, 429-437, 1973.

Bell, T. C.; Cleary, J. G.; and Witten, I. H. Text Compression. Englewood Cliffs, NJ: 1990.

Bernstein, M. and Sloane, N. J. A. "Some Canonical Sequences of Integers." Linear Algebra Appl. 226//228, 57-72, 1995.

Bower, C. G. "Further Transformations of Integer Sequences." https://www.research.att.com/~njas/sequences/transforms2.html.

Cameron, P. J. "Some Sequences of Integers." Disc. Math. 75, 89-102, 1989.

Ding, C.; Helleseth, T.; and Niederreiter, H. (Eds.). Sequences and Their Applications: Proceedings of SETA' 98. New York: Springer-Verlag, 1999.

Erdős, P.; Sárkőzy, E.; and Szemerédi, E. "On Divisibility Properties of Sequences of Integers." In Number Theory, Colloq. Math. Soc. János Bolyai, Vol. 2. Amsterdam, Netherlands: North-Holland, pp. 35-49, 1970.

Guy, R. K. "Sequences of Integers." Ch. E in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 199-239, 1994.

Kimberling, C. "Integer Sequences and Arrays." https://faculty.evansville.edu/ck6/integer/.

Krattenthaler, C. "RATE: A Mathematica Guessing Machine." https://radon.mat.univie.ac.at/People/kratt/rate/rate.html.

Loxton, J. H. "Spectral Studies of Automata." In Irregularities of Partitions (Ed. G. Halász and V. T. Sós). New York: Springer-Verlag, pp. 115-128, 1989.

Ostman, H. Additive Zahlentheorie I, II. Heidelberg, Germany: Springer-Verlag, 1956.

Pegg, E. Jr. "Math Games: Sequence Pictures." Dec. 8, 2003. https://www.maa.org/editorial/mathgames/mathgames_12_08_03.html.

Pegg, E. Jr. and Weisstein, E. W. "Seven Mathematical Tidbits." MathWorld Headline News. Nov. 8, 2004. https://mathworld.wolfram.com/news/2004-11-08/seventidbits/#3.

Peterson, I. "MathTrek: Sequence Puzzles." May 17, 2003. https://www.sciencenews.org/20030517/mathtrek.asp.

Petit, S. "Encyclopedia of Combinatorial Structures." https://algo.inria.fr/encyclopedia/.

Pomerance, C. and Sárközy, A. "Combinatorial Number Theory." In Handbook of Combinatorics (Ed. R. Graham, M. Grötschel, and L. Lovász). Amsterdam, Netherlands: North-Holland, 1994.

Ruskey, F. "The (Combinatorial) Object Server." https://www.theory.csc.uvic.ca/~cos/.

Sloane, N. J. A. A Handbook of Integer Sequences. Boston, MA: Academic Press, 1973.

Sloane, N. J. A. "Find the Next Term." J. Recr. Math. 7, 146, 1974.

Sloane, N. J. A. "An On-Line Version of the Encyclopedia of Integer Sequences." Electronic J. Combinatorics 1, No. 1, F1, 1-5, 1994. https://www.combinatorics.org/Volume_1/Abstracts/v1i1f1.html.

Sloane, N. J. A. "An On-Line Version of the Encyclopedia of Integer Sequences." https://www.research.att.com/~njas/sequences/eisonline.html.

Sloane, N. J. A. "Some Important Integer Sequences." In CRC Standard Mathematical Tables and Formulae. (Ed. D. Zwillinger). Boca Raton, FL: CRC Press, 1995.

Sloane, N. J. A. "The On-Line Encyclopedia of Integer Sequences." Not. Amer. Math. Soc. 50, 912-915, 2003.

Sloane, N. J. A. "Transformation of Integer Sequences." https://www.research.att.com/~njas/sequences/transforms.html.

Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.

Stephan, R. "Prove or Disprove. 100 Conjectures from the OEIS." 27 Sep 2004. https://www.arxiv.org/abs/math.CO/0409509/.

Stephan, R. "Do you have a comment or news on conjectures in the article math.CO/0409509?" https://www.ark.in-berlin.de/conj.txt.

Stöhr, A. "Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe I, II." J. reine angew. Math. 194, 40-65 and 111-140, 1955.

Turán, P. (Ed.). Number Theory and Analysis: A Collection of Papers in Honor of Edmund Landau (1877-1938). New York: Plenum Press, 1969.