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A194633 - OEIS
<!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>A194633 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A194633" 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=%2fA194633">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="A194633 - OEIS"></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 class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A194633 </div> <div class=seqname> Arises in enumerating Huffman codes, compact trees, and sums of unit fractions. </div> </div> <div class=scorerefs> 4 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4089, 8176, 16348, 32688, 65360, 130688, 261312, 522496, 1044736, 2088960, 4176896, 8351746, 16699401, 33390622, 66764888, 133497072, 266928752, 533726752, 1067192064, 2133861376, 4266677504, 8531265024</div> <div class=seqdatalinks> (<a href="/A194633/list">list</a>; <a href="/A194633/graph">graph</a>; <a href="/search?q=A194633+-id:A194633">refs</a>; <a href="/A194633/listen">listen</a>; <a href="/history?seq=A194633">history</a>; <a href="/search?q=id:A194633&fmt=text">text</a>; <a href="/A194633/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,4</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>a(n+1) is the number of compositions n=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 10*p(k+1). [<a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, Dec 18 2012]</div> <div class=sectline>Row 9 of Table 1 of Elsholtz, row 1 being <a href="/A002572" title="Number of partitions of 1 into n powers of 1/2; or (according to one definition of "binary") the number of binary rooted trees.">A002572</a>, row 2 being <a href="/A176485" title="First column of triangle in A176452.">A176485</a>, row 3 being <a href="/A176503" title="Leading column of triangle in A176463.">A176503</a>, row 4 being <a href="/A194628" title="Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.">A194628</a>, row 5 being <a href="/A194629" title="Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.">A194629</a>, row 6 being <a href="/A194630" title="Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.">A194630</a>, row 7 being <a href="/A194631" title="Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.">A194631</a>, and row 8 being <a href="/A194632" title="Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.">A194632</a>.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Alois P. Heinz, <a href="/A194633/b194633.txt">Table of n, a(n) for n = 1..1000</a></div> <div class=sectline>Christian Elsholtz, Clemens Heuberger, Helmut Prodinger, <a href="https://arxiv.org/abs/1108.5964">The number of Huffman codes, compact trees, and sums of unit fractions</a>, arXiv:1108.5964 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = <a href="/A294775" title="Number A(n,k) of partitions of 1 into exactly k*n+1 powers of 1/(k+1); square array A(n,k), n>=0, k>=0, read by antidiagonals.">A294775</a>(n-1,9).</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>b[n_, r_, k_] := b[n, r, k] = If[n < r, 0, If[r == 0, If[n == 0, 1, 0], Sum[b[n-j, k*(r-j), k], {j, 0, Min[n, r]}]]];</div> <div class=sectline>a[n_] := b[9n-8, 1, 10];</div> <div class=sectline>Array[a, 40] (* <a href="/wiki/User:Jean-Fran莽ois_Alcover">Jean-Fran莽ois Alcover</a>, Jul 21 2018, after <a href="/wiki/User:Alois_P._Heinz">Alois P. Heinz</a> *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) /* see <a href="/A002572" title="Number of partitions of 1 into n powers of 1/2; or (according to one definition of "binary") the number of binary rooted trees.">A002572</a>, set t=10 */</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A002572" title="Number of partitions of 1 into n powers of 1/2; or (according to one definition of "binary") the number of binary rooted trees.">A002572</a>, <a href="/A176485" title="First column of triangle in A176452.">A176485</a>, <a href="/A176503" title="Leading column of triangle in A176463.">A176503</a>, <a href="/A194628" title="Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.">A194628</a> - <a href="/A194632" title="Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.">A194632</a>, <a href="/A294775" title="Number A(n,k) of partitions of 1 into exactly k*n+1 powers of 1/(k+1); square array A(n,k), n>=0, k>=0, read by antidiagonals.">A294775</a>.</div> <div class=sectline>Sequence in context: <a href="/A234591" title="Number of binary words of length n which have no 0^b 1 1 0^a 1 0 1 0^b - matches, where a=1, b=2.">A234591</a> <a href="/A122265" title="10th-order Fibonacci numbers: a(n+1) = a(n)+...+a(n-9) with a(0) = ... = a(8) = 0, a(9) = 1.">A122265</a> <a href="/A339073" title="Number of strings of Hebrew letters with a gematria value equal to n.">A339073</a> * <a href="/A243088" title="Number of compositions of n into parts with multiplicity not larger than 10.">A243088</a> <a href="/A113699" title="Index of the occurrence of n in A113698.">A113699</a> <a href="/A113010" title="{Number of digits of n} raised to the power of {the sum of the digits of n}.">A113010</a></div> <div class=sectline>Adjacent sequences: <a href="/A194630" title="Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.">A194630</a> <a href="/A194631" title="Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.">A194631</a> <a href="/A194632" title="Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.">A194632</a> * <a href="/A194634" title="Numbers n such that k= n^2 + n + 41 is composite and there is no integer x such that n= x^2 + 40; n= (x^2+x)/2 + 81; or n= 3...">A194634</a> <a href="/A194635" title="Indices of records in A194591 restricted to prime indices.">A194635</a> <a href="/A194636" title="Least k >= 0 such that (2*n-1)*2^k - 1 or (2*n-1)*2^k + 1 is prime, or -1 if no such value exists.">A194636</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Jonathan_Vos_Post">Jonathan Vos Post</a>, Aug 30 2011</div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Added terms beyond a(20)=130688, <a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, Dec 18 2012</div> <div class=sectline>Invalid empirical g.f. removed by <a href="/wiki/User:Alois_P._Heinz">Alois P. Heinz</a>, Nov 08 2017</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </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 28 04:31 EST 2024. Contains 378181 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>