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Transformations of Integer Sequences
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>Transformations of Integer Sequences</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/transforms.html" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2ftransforms.html">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="Transformations of Integer Sequences"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div align="center"> <h1>Transformations of Integer Sequences</h1><p> A subpage of the <a href="http://oeis.org/">The On-Line Encyclopedia of Integer Sequences</a>, which makes extensive use of these transformations. </div> <p> <IMG SRC="http://www.research.att.com/~njas/banners/BluePinLeft.gif" ALT=" "> <STRONG>Keywords</STRONG>: AND-convolution, BINOMIAL transform, BINOMIALi transform, BISECT, boustrophedon transform, characteristic function, complement of sequence, compose two sequences, functional square root, continuant transform, convolution transform, inverse convolution transform, decimate, first differences, DIGREV = reverse digits, DIGSUM = sum of digits, Dirichlet convolution, Euler transform, Euleri transform, exponential transform, exponential convolution, GCD-convolution, Hankel transform, inverse of permutation, INVERT transform, INVERTi transform, LAH transform, LAHi transform, Lambert function, LCM-convolution, little Hankel transform. logarithmic transform, MOEBIUS transform, MOBIUS transform, MOEBIUSi transform, MOBIUSi transform, OR-convolution, partial products, partial sums, PARTITION transform, PARTITIONi transform, RECORDS transform, revert (or reversion), REVEGF = reversion of e.g.f., SERIESTOLISTDIV, SERIESTOLISTMULT, SERIESTOSERIESDIV, SERIESTOSERIESMULT, SERIES2, SERIES2TOLIST, SERIES2TOLISTMULT, SERIES2HTOLIST, SERIES2HTOLISTMULT, sort, STIRLING transform, STIRLINGi transform, Stirling-Bernoulli transform, SUPPORT transform, trisect, XOR-convolution, WEIGH transform, Maple, Mathematica, PARI. <p> <center><hr></center> This page has links to four other web pages which give procedures for performing a large number of useful transformations on sequences and numbers. <STRONG>The programs</STRONG><p> <ul> <li> <a href="transforms.txt"><STRONG>Maple</STRONG></a> programs. </li> <p> <li> <a href="seqtranslib.html"><STRONG>Mathematica</STRONG></a> programs, written by Olivier Gerard.<p> </li> <p> <li> <a href="transforms_pari.txt"><STRONG>PARI</STRONG></a> programs, written by Christian G. Bower. </li> <p> <li> See also the description of some <a href="transforms2.html"><STRONG>further transforms</STRONG></a> written by Christian G. Bower.<p> </li> <p> </ul> <STRONG>References</STRONG><p> <ul> <li> M. Bernstein & N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301"> Some canonical sequences of integers</a>, <i>Linear Algebra and its Applications</i>, <STRONG>226-228</STRONG> (1995), 57-72.<p> </li> <li> P. J. Cameron, Some sequences of integers, <i>Discrete Math.</i>, <STRONG>75</STRONG> (1989), 89-102.<p> </li> <li> J. Millar, N. J. A. Sloane and N. E. Young, <a href="https://arxiv.org/abs/math/0205218"> A new operation on sequences: the Boustrophedon transform,</a> J. Comb. Theory, 17A 44-54 1996.<p> </li> <li> N. J. A. Sloane and S. Plouffe, <a href="book.html">The Encyclopedia of Integer Sequences</a>, <a href="http://www.apnet.com/">Academic Press</a>, San Diego, 1995, especially Section 2.7.<p> </li> </ul><p> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified November 24 17:37 EST 2024. Contains 378083 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>