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<!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>More Transformations of Integer Sequences</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/transforms2.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=%2ftransforms2.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="More 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> <a href="/hints.html">Hints</a> </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>Further Transformations of Integer Sequences</h1> </div> This page was created by Christian G. Bower and is a sub-page of the <a href="https://oeis.org">On-Line Encyclopedia of Integer Sequences</a>. <IMG SRC="BluePinLeft.gif" ALT=" "> Keywords: AFJ, AFK, AGJ, AGK, AIJ, BFJ, BFK, BGJ, BGK, BHJ, BHK, BIJ, BIK, CFJ, CFK, CGJ, CGK, CHJ, CHK, CIJ, CIK, DFJ, DFK, DGJ, DGK, DHJ, DHK, DIJ, DIK, EFJ, EFK, EGJ. Table of Contents <ol> <li><a href="#definition">Definition.</a> </li> <li><a href="#algorithm">Algorithms.</a> </li> <li><a href="#catalogue">Catalogue of Sequences.</a> </li> <li><a href="index.html">Back to Integer Sequences.</a> </li> </ol> <hr> <a name="definition"></a> Part 1: Definition This is a generalization of transforms that count the ways objects can be partitioned. Say we have boxes of different colors and sizes. The sequence {an;n>=1} represents the number of colors a box holding n balls can be. The transformed sequence {bn;n>=1} represents the number of ways we can have a collection of boxes so that the total number of balls is n, subject to the following rules. <center><TABLE border=1 cellpadding=10 width="80%"> <caption>The boxes are ordered in one of the following ways:</caption> <tr><td> A. Linear (ordered) The boxes are in a line from beginning to end. B. Linear with turning over (reversible) The boxes are in a line that can be read in either direction. C. Circular (necklace) The boxes are in a circle. D. Circular with turning over (bracelet) The boxes are in a circle that can be read in either direction. E. None (unordered) The order of the boxes is not important. </td></tr></TABLE></center> <center><TABLE border=1 cellpadding=10 width="80%"> <caption>One of the following distinctness rules applies:</caption> <tr><td> F. Size No two boxes are the same size. G. Element No two boxes are the same size and color. H. Identity Any two boxes can be distinguished by size, color and position. I. None (indistinct) No restriction. </td></tr></TABLE></center> Distinctness H (identity) has different implications depending on the chosen order. <ul> <li>If order A is chosen, distinctness H is the same as distinctness I. </li> <li>If order B is chosen, the boxes cannot form a palindrome of length greater than one. Red 1 Blue 2 Red 1 is not allowed. </li> <li>If order C is chosen, the sequence of boxes is aperiodic. It cannot be a repitition of a shorter subsequence. Red 1 Blue 2 Red 1 Blue 2 is not allowed. </li> <li>If order D is chosen, the boxes are aperiodic and cannot be a palindrome of length greater than two. </li> <li>If order E is chosen, distinctness H is the same as distinctness G. </li> </ul> <center><TABLE border=1 cellpadding=10 width="80%"> <caption>One of the following labelling rules applies:</caption> <tr><td> J. Labeled The balls in the boxes are labeled. K. Unlabeled The balls in the boxes are not labeled. </td></tr></TABLE></center> Each transform is identified by a 3 letter code, e.g. BGJ to represent linear order with turning over, each object distinct, labeled. An X is a wild card as in CXK, unlabeled necklace transforms. AIK is the transform <a href="transforms.html">INVERT</a>. EGK is the transform <a href="transforms.html">WEIGH</a>. EIJ is the transform <a href="transforms.html">EXP</a>. EIK is the transform <a href="transforms.html">EULER</a>. There are 5×4×2=40 of these transforms. However, the AHX and EHX transforms are redundant, leaving 36. Four of them are named. As far as I know, the other 32 are not. The new and old sequence listed illustrate the 32 new transforms. Terminology: <ul> <li>XXXk means the transform XXX with exactly k boxes. These are denoted by XXX[k] in the <a href="index.html"> On-Line Encyclopedia of Integer Sequences</a>. AIK2 is the transform <a href="transforms.html">CONV</a>. </li> <li>Bracelet means necklace that can be turned over. <a href="http://www.theory.csc.uvic.ca/~cos/inf/neck/NecklaceInfo.html">More information about necklaces</a>. </li> <li>Compound windmill is a rooted planar tree where the sub-rooted tree extending from a node can be rotated independently of the rest of the tree. Much like some children's toys or carnival rides. Compound windmills can be dyslexic. </li> <li>Dyslexic planar tree is a planar tree where each sub-rooted tree extending from a node can be read from left to right or right to left. It can be thought of as viewed by an observer who does not know left from right or as sub-rooted trees that can be turned around independent of the rest of the tree. </li> <li>Eigensequence means a sequence that is stable under a given transform or is modified in some simple way. Eigensequences are covered in detail in: M. Bernstein & N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/eigen.ps"> Some canonical sequences of integers</a>, Linear Algebra and its Applications, 226-228 (1995), 57-72. <a href="A000081">A000081</a>, rooted trees, 1,1,2,4,9,20,48,115... is an eigensequence of the transform <a href="transforms.html">EULER</a>. because the transformed sequence, 1,2,4,9,20,48,115,286,..., is the original sequence shifted left one place. </li> <li>Identity bracelet means bracelet where every bead is distinguished by position and color, i.e. a bracelet generated by the transform DHK. </li> </ul> <hr> <a name="algorithm"></a> Part 2: Algorithms an is the input sequence. bn is the output sequence. A(x) is the generating function of an. B(x) is the generating function of bn. (XXX a)n = sum{k=1 to n} (XXXk a)n MÖBIUS · XXX refers to the Möbius transform of the sequence transformed by XXX. Similarly for MÖBIUS-1 · XXX. However, (MÖBIUS · XXX)k and (MÖBIUS-1 · XXX)k are defined as follows: (MÖBIUS &#183 XXX)kan = sum{d|k and d|n} (µ(d) × XXXk/dan/d) (MÖBIUS-1 &#183 XXX)kan = sum{d|k and d|n} (XXXk/dan/d) AIK = INVERT B(x) = A(x) / (1-A(x)) AIKk B(x) = A(x)k LPALk (Linear palindrome) If n, k even: bn = (AIKk/2a)n/2 If n odd, k even: bn = 0 If n even, k odd: bn = sum{i>0 and i<n/2} (a2i × (AIK(k-1)/2a)n/2-i) if n,k odd: bn = sum{i>0 and i<n/2} (a2i-1 × (AIK(k-1)/2a)(n+1)/2-i) BIKk bn = ((AIKka)n + (LPALka)n) / 2 BHKk k=1: bn = an k>1: bn = ((AIKka)n - (LPALka)n) / 2 CHKk bn = (MÖBIUS · AIK)kan / n CIK CIK = MÖBIUS-1 · CHK CPALk (Circular palindrome) CPAL1 = IDENTITY CPAL2 = CIK2 k>2: If n, k even: bn = (I+J)/2+K+L+M where: (No boxes joined) I=(AIKk/2a)n/2 (2 boxes joined are identical) J=sum{i=1 to n/2}(AIKk/2-1a)(n-2i)/2 (2 boxes joined are even and different sizes) K=sum{i,j even, j>i, i+j<n} (ai × aj × (AIKk/2-1a)(n-i-j)/2) (2 boxes joined are odd and different sizes) L=sum{i,j odd, j>i, i+j<n} (ai × aj × (AIKk/2-1a)(n-i-j)/2) (2 boxes joined are the same size and different colors) M=sum{i>0 and i<n/2} ((ai2-ai)/2 × (AIKk/2-1a)(n-2i)/2) If n odd, k even: bn = sum{i odd, j even, i+j<n} (ai × aj × ((AIKk/2-1a)(n-i-j)/2) If n even, k odd: bn = sum{i>0 and i<n/2} (a2i × (AIK(k-1)/2a)n/2-i) if n,k odd: bn = sum{i>0 and i<n/2} (a2i-1 × (AIK(k-1)/2a)(n+1)/2-i) DIKk bn = ((CIKka)n + (CPALka)n) / 2 DHKk DHK1 = IDENTITY DHK2 = CHK2 For k>2: DHKk = (MÖBIUS · (CIK - CPAL)/2)k If EXX is one of: {EFJ, EFK, EGJ, EGK, EIJ} then: AXXk = k! × EXXk BXXk = max(1,k!/2) × EXXk CXXk = (k-1!) × EXXk DXXk = max(1,(k-1)!/2) × EXXk To calculate (EFXka)n, enumerate the distinct partitions of n into k parts as terms of the following form: p1+p2+...+pk Sum the terms calculated as follows: EFJk: prod{i=1 to k}api × n! / prod{i=1 to k}pi! EFKk: prod{i=1 to k}api EFK can also be calculated as: B(x)=prod{k=1 to infinity}(1+akxk). To calculate (AIJka)n, (BHJka)n, (CHJka)n or (EGXka)n, enumerate the partitions of of n into k parts as terms of the following form: p1q1+p2q2+...+pjqj where all the pi's are distinct. Sum the terms calculated as follows: AIJk: prod{i=1 to j}apiqi × n! × k! / ((prod{i=1 to j}pi!qi) × (prod{i=1 to j}qi!)) BHJk: term1 = prod{i=1 to j}apiqi × k! / (prod{i=1 to j}qi!) term2 = prod{i=1 to j}api[qi/2] × [k/2]! / (prod{i=1 to j}[qi/2]!) If more than 1 qi is odd: term3 = term1 otherwise: term3 = term1 - term2 term = term3 × n! / prod{i=1 to j}pi!qi / 2 CHJk: term2 = sum{d|qm for all m} (µ(d) × prod{i=1 to j}api[qi/d] × [k/d]! / (prod{i=1 to j}[qi/d]!)) term = term2 × n! / prod{i=1 to j}pi!qi / k EGJk: prod{i=1 to j}C(api,qi) × n! / prod{i=1 to j}pi!qi EGKk: prod{i=1 to j}C(api,qi) DHJ: Work is in progress. <hr> <a name="catalogue"></a> Part 3: Catalogue of sequences This table identifies a formula for each sequence, usually based on one of the transforms. This should provide a convenient way to browse the sequences and see how the transforms apply to a broad class of mathematics. The base sequences: These transforms have been applied to one of the base sequences defined in the following table or to sequences in the <a href=""> On-line Encyclopedia of Integer Sequences</a>, identified by number. <TABLE cellspacing=10> <tr> <td> s1, s2, s3... </td><td> sk1 = k, skn=0 for n>1 </td> </tr><tr> <td> all1, all2, all3,... </td><td> allkn = k for all n </td> </tr><tr> <td> codd (characteristic of odd) </td><td> coddn = 1 if n is odd, 0 otherwise </td> </tr><tr> <td> noone </td><td> noone1=0, noonen=1 for n>1 </td> </tr><tr> <td> twoone </td><td> twoone1=2, twoonen=1 for n>1 </td> </tr><tr> <td> iden </td><td> idenn=n </td> </tr><tr> <td> odd </td><td> oddn=2n-1 </td> </tr><tr> <td> even </td><td> evenn=2n </td></tr></TABLE> If T is a transform: Left(n;k1, k2,..., kn)T is the <a href="http://www.research.att.com/~njas/doc/eigen.ps">eigensequence</a> that shifts left n places under T and has ai=ki for 1<=i<=n. M2(n)T is the <a href="http://www.research.att.com/~njas/doc/eigen.ps">eigensequence</a> that doubles the terms whose indices are greater than 1 under T. AFJ sequences <TABLE> <tr> <td> <a href="A032000">A032000</a> </td><td> AFJ all2 </td> </tr><tr> <td> <a href="A032001">A032001</a> </td><td> AFJ twoone </td> </tr><tr> <td> <a href="A032002">A032002</a> </td><td> AFJ iden </td> </tr><tr> <td> <a href="A032003">A032003</a> </td><td> AFJ odd </td> </tr><tr> <td> <a href="A032004">A032004</a> </td><td> Left(1;1)AFJ </td> </tr></TABLE> AFK sequences <TABLE> <tr> <td> <a href="A032005">A032005</a> </td><td> AFK all2 </td> </tr><tr> <td> <a href="A032006">A032006</a> </td><td> AFK twoone </td> </tr><tr> <td> <a href="A032007">A032007</a> </td><td> AFK iden </td> </tr><tr> <td> <a href="A032008">A032008</a> </td><td> AFK odd </td> </tr><tr> <td> <a href="A032009">A032009</a> </td><td> Left(1;1)AFK </td> </tr><tr> <td> <a href="A032010">A032010</a> </td><td> (CFK <a href="A032009">A032009</a> )n-1 </td> </tr></TABLE> AGJ sequences <TABLE> <tr> <td> <a href="A032011">A032011</a> </td><td> AGJ all1 </td> </tr> <tr> <td> <a href="A032012">A032012</a> </td><td> AGJ codd </td> </tr> <tr> <td> <a href="A032013">A032013</a> </td><td> AGJ noone </td> </tr> <tr> <td> <a href="A032014">A032014</a> </td><td> AGJ all2 </td> </tr> <tr> <td> <a href="A032015">A032015</a> </td><td> AGJ twoone </td> </tr> <tr> <td> <a href="A032016">A032016</a> </td><td> AGJ iden </td> </tr> <tr> <td> <a href="A032017">A032017</a> </td><td> AGJ odd </td> </tr> <tr> <td> <a href="A032018">A032018</a> </td><td> Left(1;1)AGJ </td> </tr> <tr> <td> <a href="A032019">A032019</a> </td><td> M2(2)AGJ </td> </tr> </TABLE> AGK sequences <TABLE> <tr> <td> <a href="A032020">A032020</a> </td><td> AGK all1 </td> </tr> <tr> <td> <a href="A032021">A032021</a> </td><td> AGK codd </td> </tr> <tr> <td> <a href="A032022">A032022</a> </td><td> AGK noone </td> </tr> <tr> <td> <a href="A032023">A032023</a> </td><td> AGK all2 </td> </tr> <tr> <td> <a href="A032024">A032024</a> </td><td> AGK twoone </td> </tr> <tr> <td> <a href="A032025">A032025</a> </td><td> AGK iden </td> </tr> <tr> <td> <a href="A032026">A032026</a> </td><td> AGK odd </td> </tr> <tr> <td> <a href="A032027">A032027</a> </td><td> Left(1;1)AGK </td> </tr> <tr> <td> <a href="A032028">A032028</a> </td><td> (CGK <a href="A032027">A032027</a> )n-1 </td> </tr> <tr> <td> <a href="A032029">A032029</a> </td><td> Left(2;1,1)AGK </td> </tr> <tr> <td> <a href="A032030">A032030</a> </td><td> M2(2)AGK </td> </tr> </TABLE> AIJ sequences <TABLE> <tr> <td> <a href="A000142">A000142</a> </td><td> AIJ s1 </td> </tr> <tr> <td> <a href="A000165">A000165</a> </td><td> AIJ s2 </td> </tr> <tr> <td> <a href="A032031">A032031</a> </td><td> AIJ s3 </td> </tr> <tr> <td> <a href="A000670">A000670</a> </td><td> AIJ all1 </td> </tr> <tr> <td> <a href="A000918">A000918</a> </td><td> AIJ2 all1 </td> </tr> <tr> <td> <a href="A001117">A001117</a> </td><td> AIJ3 all1 </td> </tr> <tr> <td> <a href="A000919">A000919</a> </td><td> AIJ4 all1 </td> </tr> <tr> <td> <a href="A001118">A001118</a> </td><td> AIJ5 all1 </td> </tr> <tr> <td> <a href="A000920">A000920</a> </td><td> AIJ6 all1 </td> </tr> <tr> <td> <a href="A006154">A006154</a> </td><td> AIJ codd </td> </tr> <tr> <td> <a href="A032032">A032032</a> </td><td> AIJ noone </td> </tr> <tr> <td> <a href="A004123">A004123</a> </td><td> AIJ all2 </td> </tr> <tr> <td> <a href="A006155">A006155</a> </td><td> AIJ twoone </td> </tr> <tr> <td> <a href="A032033">A032033</a> </td><td> AIJ all3 </td> </tr> <tr> <td> <a href="A006153">A006153</a> </td><td> AIJ iden </td> </tr> <tr> <td> <a href="A000354">A000354</a> </td><td> AIJ odd </td> </tr> <tr> <td> <a href="A001147">A001147</a> </td><td> Left(1;1)AIJ </td> </tr> <tr> <td> <a href="A032034">A032034</a> </td><td> Left(1;2)AIJ </td> </tr> <tr> <td> <a href="A032035">A032035</a> </td><td> Left(2;1,1)AIJ </td> </tr> <tr> <td> <a href="A032036">A032036</a> </td><td> Left(3;1,1,1)AIJ </td> </tr> <tr> <td> <a href="A032037">A032037</a> </td><td> M2(1)AIJ </td> </tr> </TABLE> BFJ sequences <TABLE> <tr> <td> <a href="A032038">A032038</a> </td><td> BFJ all2 </td> </tr> <tr> <td> <a href="A032039">A032039</a> </td><td> BFJ twoone </td> </tr> <tr> <td> <a href="A032040">A032040</a> </td><td> BFJ iden </td> </tr> <tr> <td> <a href="A032041">A032041</a> </td><td> BFJ odd </td> </tr> <tr> <td> <a href="A032042">A032042</a> </td><td> Left(1;1)BFJ </td> </tr> </TABLE> BFK sequences <TABLE> <tr> <td> <a href="A032043">A032043</a> </td><td> BFK all2 </td> </tr> <tr> <td> <a href="A032044">A032044</a> </td><td> BFK twoone </td> </tr> <tr> <td> <a href="A032045">A032045</a> </td><td> BFK iden </td> </tr> <tr> <td> <a href="A032046">A032046</a> </td><td> BFK odd </td> </tr> <tr> <td> <a href="A032047">A032047</a> </td><td> Left(1;1)BFK </td> </tr> <tr> <td> <a href="A032048">A032048</a> </td><td> (CFK <a href="A032047">A032047</a> )n-1 </td> </tr> </TABLE> BGJ sequences <TABLE> <tr> <td> <a href="A032049">A032049</a> </td><td> BGJ all1 </td> </tr> <tr> <td> <a href="A032050">A032050</a> </td><td> BGJ codd </td> </tr> <tr> <td> <a href="A032051">A032051</a> </td><td> BGJ noone </td> </tr> <tr> <td> <a href="A032052">A032052</a> </td><td> BGJ all2 </td> </tr> <tr> <td> <a href="A032053">A032053</a> </td><td> BGJ twoone </td> </tr> <tr> <td> <a href="A032054">A032054</a> </td><td> BGJ iden </td> </tr> <tr> <td> <a href="A032055">A032055</a> </td><td> BGJ odd </td> </tr> <tr> <td> <a href="A032056">A032056</a> </td><td> Left(1;1)BGJ </td> </tr> <tr> <td> <a href="A032057">A032057</a> </td><td> M2(2)BGJ </td> </tr> </TABLE> BGK sequences <TABLE> <tr> <td> <a href="A032058">A032058</a> </td><td> BGK all1 </td> </tr> <tr> <td> <a href="A032059">A032059</a> </td><td> BGK codd </td> </tr> <tr> <td> <a href="A032060">A032060</a> </td><td> BGK noone </td> </tr> <tr> <td> <a href="A032061">A032061</a> </td><td> BGK all2 </td> </tr> <tr> <td> <a href="A032062">A032062</a> </td><td> BGK twoone </td> </tr> <tr> <td> <a href="A032063">A032063</a> </td><td> BGK iden </td> </tr> <tr> <td> <a href="A032064">A032064</a> </td><td> BGK odd </td> </tr> <tr> <td> <a href="A032065">A032065</a> </td><td> Left(1;1)BGK </td> </tr> <tr> <td> <a href="A032066">A032066</a> </td><td> (CGK <a href="A032065">A032065</a> )n-1 </td> </tr> <tr> <td> <a href="A032067">A032067</a> </td><td> Left(2;1,1)BGK </td> </tr> <tr> <td> <a href="A032068">A032068</a> </td><td> M2(2)BGK </td> </tr> </TABLE> BHJ sequences <TABLE> <tr> <td> <a href="A032069">A032069</a> </td><td> BHJ s2 </td> </tr> <tr> <td> <a href="A032070">A032070</a> </td><td> BHJ s3 </td> </tr> <tr> <td> <a href="A032071">A032071</a> </td><td> BHJ s4 </td> </tr> <tr> <td> <a href="A032072">A032072</a> </td><td> BHJ s5 </td> </tr> <tr> <td> <a href="A032073">A032073</a> </td><td> BHJ all1 </td> </tr> <tr> <td> <a href="A032074">A032074</a> </td><td> BHJ codd </td> </tr> <tr> <td> <a href="A032075">A032075</a> </td><td> BHJ noone </td> </tr> <tr> <td> <a href="A032076">A032076</a> </td><td> BHJ all2 </td> </tr> <tr> <td> <a href="A032077">A032077</a> </td><td> BHJ twoone </td> </tr> <tr> <td> <a href="A032078">A032078</a> </td><td> BHJ all3 </td> </tr> <tr> <td> <a href="A032079">A032079</a> </td><td> BHJ iden </td> </tr> <tr> <td> <a href="A032080">A032080</a> </td><td> BHJ odd </td> </tr> <tr> <td> <a href="A032081">A032081</a> </td><td> Left(1;1)BHJ </td> </tr> <tr> <td> <a href="A032082">A032082</a> </td><td> Left(1;2)BHJ </td> </tr> <tr> <td> <a href="A032083">A032083</a> </td><td> Left(2;1,1)BHJ </td> </tr> <tr> <td> <a href="A032084">A032084</a> </td><td> M2(2)BHJ </td> </tr> </TABLE> BHK sequences <TABLE> <tr> <td> <a href="A032085">A032085</a> </td><td> BHK s2 </td> </tr> <tr> <td> <a href="A032086">A032086</a> </td><td> BHK s3 </td> </tr> <tr> <td> <a href="A032087">A032087</a> </td><td> BHK s4 </td> </tr> <tr> <td> <a href="A032088">A032088</a> </td><td> BHK s5 </td> </tr> <tr> <td> <a href="A032089">A032089</a> </td><td> BHK codd </td> </tr> <tr> <td> <a href="A032090">A032090</a> </td><td> BHK noone </td> </tr> <tr> <td> <a href="A002620">A002620</a> </td><td> (BHK3 all1)n+2 </td> </tr> <tr> <td> <a href="A006584">A006584</a> </td><td> (BHK4 all1)n+2 </td> </tr> <tr> <td> <a href="A032091">A032091</a> </td><td> BHK5 all1 </td> </tr> <tr> <td> <a href="A032092">A032092</a> </td><td> BHK6 all1 </td> </tr> <tr> <td> <a href="A032093">A032093</a> </td><td> BHK7 all1 </td> </tr> <tr> <td> <a href="A032094">A032094</a> </td><td> BHK8 all1 </td> </tr> <tr> <td> <a href="A032095">A032095</a> </td><td> (BHKn all1)2n-1 </td> </tr> <tr> <td> <a href="A032096">A032096</a> </td><td> BHK all2 </td> </tr> <tr> <td> <a href="A032097">A032097</a> </td><td> BHK twoone </td> </tr> <tr> <td> <a href="A032098">A032098</a> </td><td> BHK all3 </td> </tr> <tr> <td> <a href="A032099">A032099</a> </td><td> BHK iden </td> </tr> <tr> <td> <a href="A032100">A032100</a> </td><td> BHK odd </td> </tr> <tr> <td> <a href="A032101">A032101</a> </td><td> Left(1;1)BHK </td> </tr> <tr> <td> <a href="A032102">A032102</a> </td><td> (DHK <a href="A032101">A032101</a> )n-1 </td> </tr> <tr> <td> <a href="A032103">A032103</a> </td><td> Left(1;2)BHK </td> </tr> <tr> <td> <a href="A032104">A032104</a> </td><td> Left(1;1,1)BHK </td> </tr> <tr> <td> <a href="A032105">A032105</a> </td><td> M2(2)BHK </td> </tr> <tr> <td> <a href="A032106">A032106</a> </td><td> (BHKn all1)2n </td> </tr> </TABLE> BIJ sequences <TABLE> <tr> <td> <a href="A001710">A001710</a> </td><td> BIJ s1 </td> </tr> <tr> <td> <a href="A032107">A032107</a> </td><td> BIJ s2 </td> </tr> <tr> <td> <a href="A032108">A032108</a> </td><td> BIJ s3 </td> </tr> <tr> <td> <a href="A032109">A032109</a> </td><td> BIJ all1 </td> </tr> <tr> <td> <a href="A009568">A009568</a> </td><td> (-1)n+1 × BIJ codd </td> </tr> <tr> <td> <a href="A032110">A032110</a> </td><td> BIJ noone </td> </tr> <tr> <td> <a href="A032111">A032111</a> </td><td> BIJ all2 </td> </tr> <tr> <td> <a href="A032112">A032112</a> </td><td> BIJ twoone </td> </tr> <tr> <td> <a href="A032113">A032113</a> </td><td> BIJ all3 </td> </tr> <tr> <td> <a href="A032114">A032114</a> </td><td> BIJ iden </td> </tr> <tr> <td> <a href="A032115">A032115</a> </td><td> BIJ odd </td> </tr> <tr> <td> <a href="A032116">A032116</a> </td><td> Left(1;1)BIJ </td> </tr> <tr> <td> <a href="A032117">A032117</a> </td><td> Left(1;2)BIJ </td> </tr> <tr> <td> <a href="A032118">A032118</a> </td><td> Left(2;1,1)BIJ </td> </tr> <tr> <td> <a href="A032119">A032119</a> </td><td> M2(1)BIJ </td> </tr> </TABLE> BIK sequences <TABLE> <tr> <td> <a href="A005418">A005418</a> </td><td> (BIK s2)n-1 </td> </tr> <tr> <td> <a href="A005418">A005418</a> </td><td> BIK all1 </td> </tr> <tr> <td> <a href="A032120">A032120</a> </td><td> BIK s3 </td> </tr> <tr> <td> <a href="A032121">A032121</a> </td><td> BIK s4 </td> </tr> <tr> <td> <a href="A032122">A032122</a> </td><td> BIK s5 </td> </tr> <tr> <td> <a href="A001224">A001224</a> </td><td> (BIK codd)n+1 </td> </tr> <tr> <td> <a href="A001224">A001224</a> </td><td> (BIK noone)n+2 </td> </tr> <tr> <td> <a href="A002620">A002620</a> </td><td> (BIK3 all1)n+1 </td> </tr> <tr> <td> <a href="A005993">A005993</a> </td><td> (BIK4 all1)n+4 </td> </tr> <tr> <td> <a href="A005994">A005994</a> </td><td> (BIK5 all1)n+5 </td> </tr> <tr> <td> <a href="A005995">A005995</a> </td><td> (BIK6 all1)n+6 </td> </tr> <tr> <td> <a href="A018210">A018210</a> </td><td> (BIK7 all1)n+7 </td> </tr> <tr> <td> <a href="A018211">A018211</a> </td><td> (BIK8 all1)n+8 </td> </tr> <tr> <td> <a href="A018212">A018212</a> </td><td> (BIK9 all1)n+9 </td> </tr> <tr> <td> <a href="A018213">A018213</a> </td><td> (BIK10 all1)n+10 </td> </tr> <tr> <td> <a href="A018214">A018214</a> </td><td> (BIK11 all1)n+11 </td> </tr> <tr> <td> <a href="A032123">A032123</a> </td><td> (BIKn all1)2n-1 </td> </tr> <tr> <td> <a href="A005654">A005654</a> </td><td> (BIKn all1)2n </td> </tr> <tr> <td> <a href="A005656">A005656</a> </td><td> (BIKn-3 all1)2n-3 </td> </tr> <tr> <td> <a href="A032124">A032124</a> </td><td> BIK all2 </td> </tr> <tr> <td> <a href="A032125">A032125</a> </td><td> BIK all3 </td> </tr> <tr> <td> <a href="A005207">A005207</a> </td><td> BIK twoone </td> </tr> <tr> <td> <a href="A032126">A032126</a> </td><td> BIK iden </td> </tr> <tr> <td> <a href="A032127">A032127</a> </td><td> BIK odd </td> </tr> <tr> <td> <a href="A032128">A032128</a> </td><td> Left(1;1)BIK </td> </tr> <tr> <td> <a href="A032129">A032129</a> </td><td> (DIK <a href="A032128">A032128</a> )n-1 </td> </tr> <tr> <td> <a href="A032130">A032130</a> </td><td> Left(1;2)BIK </td> </tr> <tr> <td> <a href="A032131">A032131</a> </td><td> Left(2;1,1)BIK </td> </tr> <tr> <td> <a href="A032132">A032132</a> </td><td> M2(1)BIK </td> </tr> <tr> <td> <a href="A032133">A032133</a> </td><td> M2(2)BIK </td> </tr> </TABLE> CFJ sequences <TABLE> <tr> <td> <a href="A032134">A032134</a> </td><td> CFJ all2 </td> </tr> <tr> <td> <a href="A032135">A032135</a> </td><td> CFJ twoone </td> </tr> <tr> <td> <a href="A032136">A032136</a> </td><td> CFJ iden </td> </tr> <tr> <td> <a href="A032137">A032137</a> </td><td> CFJ odd </td> </tr> <tr> <td> <a href="A032138">A032138</a> </td><td> Left(1;1)CFJ </td> </tr> </TABLE> CFK sequences <TABLE> <tr> <td> <a href="A032139">A032139</a> </td><td> CFK all2 </td> </tr> <tr> <td> <a href="A032140">A032140</a> </td><td> CFK twoone </td> </tr> <tr> <td> <a href="A032141">A032141</a> </td><td> CFK iden </td> </tr> <tr> <td> <a href="A032142">A032142</a> </td><td> CFK odd </td> </tr> <tr> <td> <a href="A032143">A032143</a> </td><td> Left(1;1)CFK </td> </tr> </TABLE> CGJ sequences <TABLE> <tr> <td> <a href="A032144">A032144</a> </td><td> CGJ all1 </td> </tr> <tr> <td> <a href="A032145">A032145</a> </td><td> CGJ codd </td> </tr> <tr> <td> <a href="A032146">A032146</a> </td><td> CGJ noone </td> </tr> <tr> <td> <a href="A032147">A032147</a> </td><td> CGJ all2 </td> </tr> <tr> <td> <a href="A032148">A032148</a> </td><td> CGJ twoone </td> </tr> <tr> <td> <a href="A032149">A032149</a> </td><td> CGJ iden </td> </tr> <tr> <td> <a href="A032150">A032150</a> </td><td> CGJ odd </td> </tr> <tr> <td> <a href="A032151">A032151</a> </td><td> Left(1;1)CGJ </td> </tr> <tr> <td> <a href="A032152">A032152</a> </td><td> M2(2)CGJ </td> </tr> </TABLE> CGK sequences <TABLE> <tr> <td> <a href="A032153">A032153</a> </td><td> CGK all1 </td> </tr> <tr> <td> <a href="A032154">A032154</a> </td><td> CGK codd </td> </tr> <tr> <td> <a href="A032155">A032155</a> </td><td> CGK noone </td> </tr> <tr> <td> <a href="A032156">A032156</a> </td><td> CGK all2 </td> </tr> <tr> <td> <a href="A032157">A032157</a> </td><td> CGK twoone </td> </tr> <tr> <td> <a href="A032158">A032158</a> </td><td> CGK iden </td> </tr> <tr> <td> <a href="A032159">A032159</a> </td><td> CGK odd </td> </tr> <tr> <td> <a href="A032160">A032160</a> </td><td> Left(1;1)CGK </td> </tr> <tr> <td> <a href="A032161">A032161</a> </td><td> Left(1;2)CGK </td> </tr> <tr> <td> <a href="A032162">A032162</a> </td><td> Left(2;1,1)CGK </td> </tr> <tr> <td> <a href="A032163">A032163</a> </td><td> M2(2)CGK </td> </tr> </TABLE> CHJ sequences <TABLE> <tr> <td> <a href="A032321">A032321</a> </td><td> CHJ s2 </td> </tr> <tr> <td> <a href="A032322">A032322</a> </td><td> CHJ s3 </td> </tr> <tr> <td> <a href="A032323">A032323</a> </td><td> CHJ s4 </td> </tr> <tr> <td> <a href="A032324">A032324</a> </td><td> CHJ s5 </td> </tr> <tr> <td> <a href="A032325">A032325</a> </td><td> CHJ all1 </td> </tr> <tr> <td> <a href="A032326">A032326</a> </td><td> CHJ codd </td> </tr> <tr> <td> <a href="A032327">A032327</a> </td><td> CHJ noone </td> </tr> <tr> <td> <a href="A032328">A032328</a> </td><td> CHJ all2 </td> </tr> <tr> <td> <a href="A032329">A032329</a> </td><td> CHJ twoone </td> </tr> <tr> <td> <a href="A032330">A032330</a> </td><td> CHJ all3 </td> </tr> <tr> <td> <a href="A032331">A032331</a> </td><td> CHJ iden </td> </tr> <tr> <td> <a href="A032332">A032332</a> </td><td> CHJ odd </td> </tr> <tr> <td> <a href="A032333">A032333</a> </td><td> Left(1;1)CHJ </td> </tr> <tr> <td> <a href="A032334">A032334</a> </td><td> Left(1;2)CHJ </td> </tr> <tr> <td> <a href="A032335">A032335</a> </td><td> Left(2;1,1)CHJ </td> </tr> <tr> <td> <a href="A032336">A032336</a> </td><td> M2(2)CHJ </td> </tr> </TABLE> CHK sequences <TABLE> <tr> <td> <a href="A001037">A001037</a> </td><td> CHK s2 </td> </tr> <tr> <td> <a href="A001037">A001037</a> </td><td> (CHK all1) + s1 </td> </tr> <tr> <td> <a href="A027376">A027376</a> </td><td> CHK s3 </td> </tr> <tr> <td> <a href="A027376">A027376</a> </td><td> (CHK all2) + s1 </td> </tr> <tr> <td> <a href="A027376">A027376</a> </td><td> (CHK odd) + s2 </td> </tr> <tr> <td> <a href="A027377">A027377</a> </td><td> CHK s4 </td> </tr> <tr> <td> <a href="A027377">A027377</a> </td><td> (CHK all3) + s1 </td> </tr> <tr> <td> <a href="A001692">A001692</a> </td><td> CHK s5 </td> </tr> <tr> <td> <a href="A027378">A027378</a> </td><td> CHK s5 </td> </tr> <tr> <td> <a href="A032164">A032164</a> </td><td> CHK s6 </td> </tr> <tr> <td> <a href="A001693">A001693</a> </td><td> CHK s7 </td> </tr> <tr> <td> <a href="A027379">A027379</a> </td><td> CHK s7 </td> </tr> <tr> <td> <a href="A027380">A027380</a> </td><td> CHK s8 </td> </tr> <tr> <td> <a href="A027381">A027381</a> </td><td> CHK s9 </td> </tr> <tr> <td> <a href="A032165">A032165</a> </td><td> CHK s10 </td> </tr> <tr> <td> <a href="A032166">A032166</a> </td><td> CHK s11 </td> </tr> <tr> <td> <a href="A032167">A032167</a> </td><td> CHK s12 </td> </tr> <tr> <td> <a href="A006206">A006206</a> </td><td> (CHK codd) + CHAR({2}) </td> </tr> <tr> <td> <a href="A006206">A006206</a> </td><td> (CHK noone) + s1 </td> </tr> <tr> <td> <a href="A001840">A001840</a> </td><td> (CHK3 all1)n+4 </td> </tr> <tr> <td> <a href="A006918">A006918</a> </td><td> (CHK4 all1)n+4 </td> </tr> <tr> <td> <a href="A011795">A011795</a> </td><td> (CHK5 all1)n+1 </td> </tr> <tr> <td> <a href="A011796">A011796</a> </td><td> (CHK6 all1)n+6 </td> </tr> <tr> <td> <a href="A011797">A011797</a> </td><td> (CHK7 all1)n+1 </td> </tr> <tr> <td> <a href="A031164">A031164</a> </td><td> (CHK8 all1)n+9 </td> </tr> <tr> <td> <a href="A011845">A011845</a> </td><td> CHK9 all1 </td> </tr> <tr> <td> <a href="A032168">A032168</a> </td><td> CHK10 all1 </td> </tr> <tr> <td> <a href="A032169">A032169</a> </td><td> CHK11 all1 </td> </tr> <tr> <td> <a href="A000108">A000108</a> </td><td> (CHKn+1 all1)2n+1 </td> </tr> <tr> <td> <a href="A022553">A022553</a> </td><td> (CHKn+1 all1)2n+2 </td> </tr> <tr> <td> <a href="A022553">A022553</a> </td><td> (CHK <a href="A000108">A000108</a> )n-1 </td> </tr> <tr> <td> <a href="A032170">A032170</a> </td><td> CHK iden </td> </tr> <tr> <td> <a href="A032170">A032170</a> </td><td> CHK twoone + s1 </td> </tr> <tr> <td> <a href="A032171">A032171</a> </td><td> Left(1;1)CHK </td> </tr> <tr> <td> <a href="A032172">A032172</a> </td><td> Left(1;2)CHK </td> </tr> <tr> <td> <a href="A032173">A032173</a> </td><td> Left(2;1,1)CHK </td> </tr> <tr> <td> <a href="A032174">A032174</a> </td><td> M2(2)CHK </td> </tr> <tr> <td> <a href="A032175">A032175</a> </td><td> CHK <a href="A004111">A004111</a> </td> </tr> <tr> <td> <a href="A032176">A032176</a> </td><td> WEIGH <a href="A032175">A032175</a> </td> </tr> <tr> <td> <a href="A032177">A032177</a> </td><td> <a href="A032176">A032176</a> - <a href="A004111">A004111</a> </td> </tr> <tr> <td> <a href="A032178">A032178</a> </td><td> WEIGH <a href="A032177">A032177</a> </td> </tr> </TABLE> CIJ sequences <TABLE> <tr> <td> <a href="A000142">A000142</a> </td><td> (CIJ s1)n+1 </td> </tr> <tr> <td> <a href="A000165">A000165</a> </td><td> (CIJ s2)n+1 × 2 </td> </tr> <tr> <td> <a href="A032179">A032179</a> </td><td> CIJ s3 </td> </tr> <tr> <td> <a href="A000629">A000629</a> </td><td> CIJ all1 </td> </tr> <tr> <td> <a href="A000225">A000225</a> </td><td> (CIJ2 all1)n+1 </td> </tr> <tr> <td> <a href="A028243">A028243</a> </td><td> CIJ3 all1 </td> </tr> <tr> <td> <a href="A028244">A028244</a> </td><td> CIJ4 all1 </td> </tr> <tr> <td> <a href="A028245">A028245</a> </td><td> CIJ5 all1 </td> </tr> <tr> <td> <a href="A032180">A032180</a> </td><td> CIJ6 all1 </td> </tr> <tr> <td> <a href="A003704">A003704</a> </td><td> (-1)n+1 × (CIJ codd) </td> </tr> <tr> <td> <a href="A032181">A032181</a> </td><td> CIJ noone </td> </tr> <tr> <td> <a href="A027882">A027882</a> </td><td> CIJ all2 </td> </tr> <tr> <td> <a href="A032182">A032182</a> </td><td> CIJ twoone </td> </tr> <tr> <td> <a href="A032183">A032183</a> </td><td> CIJ all3 </td> </tr> <tr> <td> <a href="A009444">A009444</a> </td><td> (-1)n+1 × (CIJ iden) </td> </tr> <tr> <td> <a href="A032184">A032184</a> </td><td> CIJ odd </td> </tr> <tr> <td> <a href="A029768">A029768</a> </td><td> Left(1;1)CIJ </td> </tr> <tr> <td> <a href="A032185">A032185</a> </td><td> Left(1;2)CIJ </td> </tr> <tr> <td> <a href="A032186">A032186</a> </td><td> Left(2;1,1)CIJ </td> </tr> <tr> <td> <a href="A032187">A032187</a> </td><td> Left(3;1,1,1)CIJ </td> </tr> <tr> <td> <a href="A032188">A032188</a> </td><td> M2(1)CIJ </td> </tr> </TABLE> CIK sequences <TABLE> <tr> <td> <a href="A000031">A000031</a> </td><td> CIK s2 </td> </tr> <tr> <td> <a href="A000031">A000031</a> </td><td> (CIK all1) + all1 </td> </tr> <tr> <td> <a href="A008965">A008965</a> </td><td> CIK all1 </td> </tr> <tr> <td> <a href="A008965">A008965</a> </td><td> (CIK s2) - all1 </td> </tr> <tr> <td> <a href="A001867">A001867</a> </td><td> CIK s3 </td> </tr> <tr> <td> <a href="A001867">A001867</a> </td><td> (CIK all2) + all1 </td> </tr> <tr> <td> <a href="A001868">A001868</a> </td><td> CIK s4 </td> </tr> <tr> <td> <a href="A001868">A001868</a> </td><td> (CIK all3) + all1 </td> </tr> <tr> <td> <a href="A001869">A001869</a> </td><td> CIK s5 </td> </tr> <tr> <td> <a href="A001869">A001869</a> </td><td> (CIK all4) + all1 </td> </tr> <tr> <td> <a href="A032189">A032189</a> </td><td> CIK codd </td> </tr> <tr> <td> <a href="A032190">A032190</a> </td><td> CIK noone </td> </tr> <tr> <td> <a href="A000358">A000358</a> </td><td> (CIK noone) + all1 </td> </tr> <tr> <td> <a href="A007997">A007997</a> </td><td> (CIK3 all1)n+3 </td> </tr> <tr> <td> <a href="A008610">A008610</a> </td><td> (CIK4 all1)n+4 </td> </tr> <tr> <td> <a href="A008646">A008646</a> </td><td> (CIK5 all1)n+5 </td> </tr> <tr> <td> <a href="A032191">A032191</a> </td><td> CIK6 all1 </td> </tr> <tr> <td> <a href="A032192">A032192</a> </td><td> CIK7 all1 </td> </tr> <tr> <td> <a href="A032193">A032193</a> </td><td> CIK8 all1 </td> </tr> <tr> <td> <a href="A032194">A032194</a> </td><td> CIK9 all1 </td> </tr> <tr> <td> <a href="A032195">A032195</a> </td><td> CIK10 all1 </td> </tr> <tr> <td> <a href="A032196">A032196</a> </td><td> CIK11 all1 </td> </tr> <tr> <td> <a href="A032197">A032197</a> </td><td> CIK12 all1 </td> </tr> <tr> <td> <a href="A000108">A000108</a> </td><td> (CHKn+1 all1)2n+1 </td> </tr> <tr> <td> <a href="A003239">A003239</a> </td><td> (CIKn-1 all1)2n-2 </td> </tr> <tr> <td> <a href="A003239">A003239</a> </td><td> (CIK <a href="A000108">A000108</a> n-1)n-1 </td> </tr> <tr> <td> <a href="A005594">A005594</a> </td><td> CIK twoone </td> </tr> <tr> <td> <a href="A032198">A032198</a> </td><td> CIK iden </td> </tr> <tr> <td> <a href="A032199">A032199</a> </td><td> CIK odd </td> </tr> <tr> <td> <a href="A032200">A032200</a> </td><td> Left(1;1)CIK </td> </tr> <tr> <td> <a href="A032201">A032201</a> </td><td> Left(1;2)CIK </td> </tr> <tr> <td> <a href="A032202">A032202</a> </td><td> Left(2;1,1)CIK </td> </tr> <tr> <td> <a href="A032203">A032203</a> </td><td> M2(1)CIK </td> </tr> <tr> <td> <a href="A032204">A032204</a> </td><td> M2(2)CIK </td> </tr> <tr> <td> <a href="A002861">A002861</a> </td><td> CIK <a href="A000081">A000081</a> </td> </tr> <tr> <td> <a href="A027852">A027852</a> </td><td> CIK2 <a href="A000081">A000081</a> </td> </tr> <tr> <td> <a href="A029852">A029852</a> </td><td> CIK3 <a href="A000081">A000081</a> </td> </tr> <tr> <td> <a href="A029853">A029853</a> </td><td> CIK4 <a href="A000081">A000081</a> </td> </tr> <tr> <td> <a href="A029868">A029868</a> </td><td> CIK5 <a href="A000081">A000081</a> </td> </tr> <tr> <td> <a href="A029869">A029869</a> </td><td> CIK6 <a href="A000081">A000081</a> </td> </tr> <tr> <td> <a href="A029870">A029870</a> </td><td> CIK7 <a href="A000081">A000081</a> </td> </tr> <tr> <td> <a href="A029871">A029871</a> </td><td> CIK8 <a href="A000081">A000081</a> </td> </tr> <tr> <td> <a href="A032205">A032205</a> </td><td> CIK9 <a href="A000081">A000081</a> </td> </tr> <tr> <td> <a href="A032206">A032206</a> </td><td> CIK10 <a href="A000081">A000081</a> </td> </tr> <tr> <td> <a href="A032207">A032207</a> </td><td> CIK11 <a href="A000081">A000081</a> </td> </tr> <tr> <td> <a href="A032208">A032208</a> </td><td> CIK12 <a href="A000081">A000081</a> </td> </tr> </TABLE> DFJ sequences <TABLE> <tr> <td> <a href="A032209">A032209</a> </td><td> DFJ all2 </td> </tr> <tr> <td> <a href="A032210">A032210</a> </td><td> DFJ twoone </td> </tr> <tr> <td> <a href="A032211">A032211</a> </td><td> DFJ iden </td> </tr> <tr> <td> <a href="A032212">A032212</a> </td><td> DFJ odd </td> </tr> <tr> <td> <a href="A032213">A032213</a> </td><td> Left(1;1)DFJ </td> </tr> </TABLE> DFK sequences <TABLE> <tr> <td> <a href="A032214">A032214</a> </td><td> DFK all2 </td> </tr> <tr> <td> <a href="A032215">A032215</a> </td><td> DFK twoone </td> </tr> <tr> <td> <a href="A032216">A032216</a> </td><td> DFK iden </td> </tr> <tr> <td> <a href="A032217">A032217</a> </td><td> DFK odd </td> </tr> <tr> <td> <a href="A032218">A032218</a> </td><td> Left(1;1)DFK </td> </tr> </TABLE> DGJ sequences <TABLE> <tr> <td> <a href="A032219">A032219</a> </td><td> DGJ all1 </td> </tr> <tr> <td> <a href="A032220">A032220</a> </td><td> DGJ codd </td> </tr> <tr> <td> <a href="A032221">A032221</a> </td><td> DGJ noone </td> </tr> <tr> <td> <a href="A032222">A032222</a> </td><td> DGJ all2 </td> </tr> <tr> <td> <a href="A032223">A032223</a> </td><td> DGJ twoone </td> </tr> <tr> <td> <a href="A032224">A032224</a> </td><td> DGJ iden </td> </tr> <tr> <td> <a href="A032225">A032225</a> </td><td> DGJ odd </td> </tr> <tr> <td> <a href="A032226">A032226</a> </td><td> Left(1;1)DGJ </td> </tr> <tr> <td> <a href="A032227">A032227</a> </td><td> M2(2)DGJ </td> </tr> </TABLE> DGK sequences <TABLE> <tr> <td> <a href="A032228">A032228</a> </td><td> DGK all1 </td> </tr> <tr> <td> <a href="A032229">A032229</a> </td><td> DGK codd </td> </tr> <tr> <td> <a href="A032230">A032230</a> </td><td> DGK noone </td> </tr> <tr> <td> <a href="A032231">A032231</a> </td><td> DGK all2 </td> </tr> <tr> <td> <a href="A032232">A032232</a> </td><td> DGK twoone </td> </tr> <tr> <td> <a href="A032233">A032233</a> </td><td> DGK iden </td> </tr> <tr> <td> <a href="A032234">A032234</a> </td><td> DGK odd </td> </tr> <tr> <td> <a href="A032235">A032235</a> </td><td> Left(1;1)DGK </td> </tr> <tr> <td> <a href="A032236">A032236</a> </td><td> Left(1;2)DGK </td> </tr> <tr> <td> <a href="A032237">A032237</a> </td><td> Left(2;1,1)DGK </td> </tr> <tr> <td> <a href="A032238">A032238</a> </td><td> M2(2)DGK </td> </tr> </TABLE> DHJ sequences <TABLE> <tr> <td> <a href="A032337">A032337</a> </td><td> DHJ s2 </td> </tr> <tr> <td> <a href="A032338">A032338</a> </td><td> DHJ s3 </td> </tr> <tr> <td> <a href="A032339">A032339</a> </td><td> DHJ s4 </td> </tr> <tr> <td> <a href="A032340">A032340</a> </td><td> DHJ s5 </td> </tr> </TABLE> DHK sequences <TABLE> <tr> <td> <a href="A032239">A032239</a> </td><td> DHK s2 </td> </tr> <tr> <td> <a href="A032240">A032240</a> </td><td> DHK s3 </td> </tr> <tr> <td> <a href="A032241">A032241</a> </td><td> DHK s4 </td> </tr> <tr> <td> <a href="A032242">A032242</a> </td><td> DHK s5 </td> </tr> <tr> <td> <a href="A032243">A032243</a> </td><td> DHK codd </td> </tr> <tr> <td> <a href="A032244">A032244</a> </td><td> DHK noone </td> </tr> <tr> <td> <a href="A032245">A032245</a> </td><td> DHK all1 </td> </tr> <tr> <td> <a href="A001399">A001399</a> </td><td> (DHK3 all1)n+6 </td> </tr> <tr> <td> <a href="A018845">A018845</a> </td><td> (DHK3 all1)n+6 </td> </tr> <tr> <td> <a href="A026809">A026809</a> </td><td> (DHK3 all1)n+3 </td> </tr> <tr> <td> <a href="A008804">A008804</a> </td><td> (DHK4 all1)n+7 </td> </tr> <tr> <td> <a href="A032246">A032246</a> </td><td> DHK5 all1 </td> </tr> <tr> <td> <a href="A032247">A032247</a> </td><td> DHK6 all1 </td> </tr> <tr> <td> <a href="A032248">A032248</a> </td><td> DHK7 all1 </td> </tr> <tr> <td> <a href="A032249">A032249</a> </td><td> DHK8 all1 </td> </tr> <tr> <td> <a href="A032250">A032250</a> </td><td> (DHKn all1)2n </td> </tr> <tr> <td> <a href="A032251">A032251</a> </td><td> DHK all2 </td> </tr> <tr> <td> <a href="A032252">A032252</a> </td><td> DHK twoone </td> </tr> <tr> <td> <a href="A032253">A032253</a> </td><td> DHK all3 </td> </tr> <tr> <td> <a href="A032254">A032254</a> </td><td> DHK iden </td> </tr> <tr> <td> <a href="A032255">A032255</a> </td><td> DHK odd </td> </tr> <tr> <td> <a href="A032256">A032256</a> </td><td> Left(1;1)DHK </td> </tr> <tr> <td> <a href="A032257">A032257</a> </td><td> Left(1;2)DHK </td> </tr> <tr> <td> <a href="A032258">A032258</a> </td><td> Left(2;1,1)DHK </td> </tr> <tr> <td> <a href="A032259">A032259</a> </td><td> M2(2)DHK </td> </tr> <tr> <td> <a href="A032260">A032260</a> </td><td> (DHKn all1)2n-1 </td> </tr> </TABLE> DIJ sequences <TABLE> <tr> <td> <a href="A001710">A001710</a> </td><td> (DIJ s1)n+1 </td> </tr> <tr> <td> <a href="A000165">A000165</a> </td><td> (DIJ s2)n+1 - s2 </td> </tr> <tr> <td> <a href="A032261">A032261</a> </td><td> DIJ s3 </td> </tr> <tr> <td> <a href="A032262">A032262</a> </td><td> DIJ all1 </td> </tr> <tr> <td> <a href="A000225">A000225</a> </td><td> (DIJ2 all1)n+1 </td> </tr> <tr> <td> <a href="A000392">A000392</a> </td><td> DIJ3 all1 </td> </tr> <tr> <td> <a href="A032263">A032263</a> </td><td> DIJ4 all1 </td> </tr> <tr> <td> <a href="A032264">A032264</a> </td><td> DIJ codd </td> </tr> <tr> <td> <a href="A032265">A032265</a> </td><td> DIJ noone </td> </tr> <tr> <td> <a href="A032266">A032266</a> </td><td> DIJ all2 </td> </tr> <tr> <td> <a href="A032267">A032267</a> </td><td> DIJ twoone </td> </tr> <tr> <td> <a href="A032268">A032268</a> </td><td> DIJ all3 </td> </tr> <tr> <td> <a href="A032269">A032269</a> </td><td> DIJ iden </td> </tr> <tr> <td> <a href="A032270">A032270</a> </td><td> DIJ odd </td> </tr> <tr> <td> <a href="A032271">A032271</a> </td><td> Left(1;1)DIJ </td> </tr> <tr> <td> <a href="A032272">A032272</a> </td><td> Left(1;2)DIJ </td> </tr> <tr> <td> <a href="A032273">A032273</a> </td><td> Left(2;1,1)DIJ </td> </tr> <tr> <td> <a href="A032274">A032274</a> </td><td> M2(1)DIJ </td> </tr> </TABLE> DIK sequences <TABLE> <tr> <td> <a href="A000029">A000029</a> </td><td> DIK s2 </td> </tr> <tr> <td> <a href="A000029">A000029</a> </td><td> (DIK all1) + all1 </td> </tr> <tr> <td> <a href="A027671">A027671</a> </td><td> DIK s3 </td> </tr> <tr> <td> <a href="A032275">A032275</a> </td><td> DIK s4 </td> </tr> <tr> <td> <a href="A032276">A032276</a> </td><td> DIK s5 </td> </tr> <tr> <td> <a href="A032277">A032277</a> </td><td> DIK codd </td> </tr> <tr> <td> <a href="A032278">A032278</a> </td><td> DIK noone </td> </tr> <tr> <td> <a href="A001399">A001399</a> </td><td> (DIK3 all1)n+3 </td> </tr> <tr> <td> <a href="A018845">A018845</a> </td><td> (DIK3 all1)n+3 </td> </tr> <tr> <td> <a href="A026809">A026809</a> </td><td> DIK3 all1 </td> </tr> <tr> <td> <a href="A005232">A005232</a> </td><td> DIK4 all1 </td> </tr> <tr> <td> <a href="A032279">A032279</a> </td><td> DIK5 all1 </td> </tr> <tr> <td> <a href="A005513">A005513</a> </td><td> DIK6 all1 </td> </tr> <tr> <td> <a href="A032280">A032280</a> </td><td> DIK7 all1 </td> </tr> <tr> <td> <a href="A005514">A005514</a> </td><td> DIK8 all1 </td> </tr> <tr> <td> <a href="A032281">A032281</a> </td><td> DIK9 all1 </td> </tr> <tr> <td> <a href="A005515">A005515</a> </td><td> DIK10 all1 </td> </tr> <tr> <td> <a href="A032282">A032282</a> </td><td> DIK11 all1 </td> </tr> <tr> <td> <a href="A005516">A005516</a> </td><td> DIK12 all1 </td> </tr> <tr> <td> <a href="A005648">A005648</a> </td><td> (DIKn all1)2n </td> </tr> <tr> <td> <a href="A007123">A007123</a> </td><td> (DIKn all1)2n-1 </td> </tr> <tr> <td> <a href="A032283">A032283</a> </td><td> DIK all2 </td> </tr> <tr> <td> <a href="A032284">A032284</a> </td><td> DIK all3 </td> </tr> <tr> <td> <a href="A032285">A032285</a> </td><td> DIK all4 </td> </tr> <tr> <td> <a href="A032286">A032286</a> </td><td> DIK all5 </td> </tr> <tr> <td> <a href="A005595">A005595</a> </td><td> DIK twoone </td> </tr> <tr> <td> <a href="A032287">A032287</a> </td><td> DIK iden </td> </tr> <tr> <td> <a href="A032288">A032288</a> </td><td> DIK odd </td> </tr> <tr> <td> <a href="A032289">A032289</a> </td><td> Left(1;1)DIK </td> </tr> <tr> <td> <a href="A032290">A032290</a> </td><td> Left(1;2)DIK </td> </tr> <tr> <td> <a href="A032291">A032291</a> </td><td> Left(2;1,1)DIK </td> </tr> <tr> <td> <a href="A032292">A032292</a> </td><td> M2(1)DIK </td> </tr> <tr> <td> <a href="A032293">A032293</a> </td><td> M2(2)DIK </td> </tr> <tr> <td> <a href="A001371">A001371</a> </td><td> MÖBIUS <a href="A000029">A000029</a> </td> </tr> <tr> <td> <a href="A032294">A032294</a> </td><td> MÖBIUS <a href="A027671">A027671</a> </td> </tr> <tr> <td> <a href="A032295">A032295</a> </td><td> MÖBIUS <a href="A032275">A032275</a> </td> </tr> <tr> <td> <a href="A032296">A032296</a> </td><td> MÖBIUS <a href="A032276">A032276</a> </td> </tr> </TABLE> EFJ sequences <TABLE> <tr> <td> <a href="A032297">A032297</a> </td><td> EFJ all2 </td> </tr> <tr> <td> <a href="A032298">A032298</a> </td><td> EFJ twoone </td> </tr> <tr> <td> <a href="A032299">A032299</a> </td><td> EFJ iden </td> </tr> <tr> <td> <a href="A032300">A032300</a> </td><td> EFJ odd </td> </tr> <tr> <td> <a href="A032301">A032301</a> </td><td> Left(1;1)EFJ </td> </tr> </TABLE> EFK sequences <TABLE> <tr> <td> <a href="A032302">A032302</a> </td><td> EFK all2 </td> </tr> <tr> <td> <a href="A032303">A032303</a> </td><td> EFK twoone </td> </tr> <tr> <td> <a href="A022629">A022629</a> </td><td> EFK iden </td> </tr> <tr> <td> <a href="A032304">A032304</a> </td><td> EFK odd </td> </tr> <tr> <td> <a href="A032305">A032305</a> </td><td> Left(1;1)EFK </td> </tr> <tr> <td> <a href="A032306">A032306</a> </td><td> Left(1;2)EFK </td> </tr> <tr> <td> <a href="A032307">A032307</a> </td><td> Left(2;1,1)EFK </td> </tr> <tr> <td> <a href="A032308">A032308</a> </td><td> EFK all3 </td> </tr> <tr> <td> <a href="A032309">A032309</a> </td><td> EFK even </td> </tr> </TABLE> EGJ sequences <TABLE> <tr> <td> <a href="A007837">A007837</a> </td><td> EGJ all1 </td> </tr> <tr> <td> <a href="A032310">A032310</a> </td><td> EGJ codd </td> </tr> <tr> <td> <a href="A032311">A032311</a> </td><td> EGJ noone </td> </tr> <tr> <td> <a href="A032312">A032312</a> </td><td> EGJ all2 </td> </tr> <tr> <td> <a href="A032313">A032313</a> </td><td> EGJ twone </td> </tr> <tr> <td> <a href="A032314">A032314</a> </td><td> EGJ all3 </td> </tr> <tr> <td> <a href="A032315">A032315</a> </td><td> EGJ iden </td> </tr> <tr> <td> <a href="A032316">A032316</a> </td><td> EGJ odd </td> </tr> <tr> <td> <a href="A032317">A032317</a> </td><td> Left(1;1)EGJ </td> </tr> <tr> <td> <a href="A032318">A032318</a> </td><td> Left(1;2)EGJ </td> </tr> <tr> <td> <a href="A032319">A032319</a> </td><td> Left(2;1,1)EGJ </td> </tr> <tr> <td> <a href="A032320">A032320</a> </td><td> M2(2)EGJ </td> </tr> </TABLE> </div></div> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome">Welcome</a> <a href="/wiki/Main_Page">Wiki</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> 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