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A176503 - 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>A176503 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A176503" 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=%2fA176503">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="A176503 - 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> A176503 </div> <div class=seqname> Leading column of triangle in <a href="/A176463" title="Irregular triangle read by rows: T(n,k) = number of Huffman-equivalence classes of ternary trees with 3n+1 leaves and 4k lea...">A176463</a>. </div> </div> <div class=scorerefs> 11 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 1, 1, 2, 4, 8, 15, 29, 57, 112, 220, 432, 848, 1666, 3273, 6430, 12632, 24816, 48754, 95783, 188177, 369696, 726312, 1426930, 2803381, 5507590, 10820345, 21257915, 41763825, 82050242, 161197933, 316693445, 622183778, 1222357651, 2401474098, 4717995460</div> <div class=seqdatalinks> (<a href="/A176503/list">list</a>; <a href="/A176503/graph">graph</a>; <a href="/search?q=A176503+-id:A176503">refs</a>; <a href="/A176503/listen">listen</a>; <a href="/history?seq=A176503">history</a>; <a href="/search?q=id:A176503&fmt=text">text</a>; <a href="/A176503/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) <= 4*p(k+1), see example. [<a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, Dec 18 2012]</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Alois P. Heinz, <a href="/A176503/b176503.txt">Table of n, a(n) for n = 1..1000</a></div> <div class=sectline>Christian Elsholtz, Clemens Heuberger, Daniel Krenn, <a href="https://arxiv.org/abs/1901.11343">Algorithmic counting of nonequivalent compact Huffman codes</a>, arXiv:1901.11343 [math.CO], 2019.</div> <div class=sectline>Christian Elsholtz, Clemens Heuberger, Helmut Prodinger, <a href="http://arxiv.org/abs/1108.5964">The number of Huffman codes, compact trees, and sums of unit fractions</a>, arXiv:1108.5964v1 [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-1,3). - <a href="/wiki/User:Alois_P._Heinz">Alois P. Heinz</a>, Nov 08 2017</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>From <a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, Dec 18 2012: (Start)</div> <div class=sectline>There are a(6+1)=15 compositions 6=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 4*p(k+1):</div> <div class=sectline>[ 1] [ 1 1 1 1 1 1 ]</div> <div class=sectline>[ 2] [ 1 1 1 1 2 ]</div> <div class=sectline>[ 3] [ 1 1 1 2 1 ]</div> <div class=sectline>[ 4] [ 1 1 1 3 ]</div> <div class=sectline>[ 5] [ 1 1 2 1 1 ]</div> <div class=sectline>[ 6] [ 1 1 2 2 ]</div> <div class=sectline>[ 7] [ 1 1 3 1 ]</div> <div class=sectline>[ 8] [ 1 1 4 ]</div> <div class=sectline>[ 9] [ 1 2 1 1 1 ]</div> <div class=sectline>[10] [ 1 2 1 2 ]</div> <div class=sectline>[11] [ 1 2 2 1 ]</div> <div class=sectline>[12] [ 1 2 3 ]</div> <div class=sectline>[13] [ 1 3 1 1 ]</div> <div class=sectline>[14] [ 1 3 2 ]</div> <div class=sectline>[15] [ 1 4 1 ]</div> <div class=sectline>(End)</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[3n-2, 1, 4];</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)</div> <div class=sectline>/* g.f. as given in the Elsholtz/Heuberger/Prodinger reference */</div> <div class=sectline>N=66; q='q+O('q^N);</div> <div class=sectline>t=4; /* t-ary: t=2 for <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>, t=3 for <a href="/A176485" title="First column of triangle in A176452.">A176485</a>, t=4 for <a href="/A176503" title="Leading column of triangle in A176463.">A176503</a> */</div> <div class=sectline>L=2 + 2*ceil( log(N) / log(t) );</div> <div class=sectline>f(k) = (1-t^k)/(1-t);</div> <div class=sectline>la(j) = prod(i=1, j, q^f(i) / ( 1 - q^f(i) ) );</div> <div class=sectline>nm=sum(j=0, L, (-1)^j * q^f(j) * la(j) );</div> <div class=sectline>dn=sum(j=0, L, (-1)^j * la(j) );</div> <div class=sectline>gf = nm / dn;</div> <div class=sectline>Vec( gf )</div> <div class=sectline>/* <a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, Dec 27 2012 */</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A176463" title="Irregular triangle read by rows: T(n,k) = number of Huffman-equivalence classes of ternary trees with 3n+1 leaves and 4k lea...">A176463</a>, <a href="/A194628" title="Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.">A194628</a> - <a href="/A194633" title="Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.">A194633</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="/A239555" title="Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-5,...,5}.">A239555</a> <a href="/A275544" title="Number of distinct terms at a given iteration of the Collatz (or 3x+1) map starting with 0.">A275544</a> <a href="/A000078" title="Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) for n >= 4 with a(0) = a(1) = a(2) = 0 and a(3) = 1.">A000078</a> * <a href="/A262333" title="Number of (n+3) X (1+3) 0..1 arrays with each row and column divisible by 9, read as a binary number with top and left being...">A262333</a> <a href="/A293335" title="Least integer k such that k/2^n > sqrt(1/5).">A293335</a> <a href="/A344687" title="a(n) is the lowest nonnegative exponent k such that n!^k is the product of the divisors of n!.">A344687</a></div> <div class=sectline>Adjacent sequences: <a href="/A176500" title="a(n) = 2*Farey(Fibonacci(n + 1)) - 3.">A176500</a> <a href="/A176501" title="a(n) = Farey(m; I) where m = Fibonacci (n + 1) and I = [1/n, 1].">A176501</a> <a href="/A176502" title="a(n) = 2*Farey(m; I) - 1 where m = Fibonacci (n + 1) and I = [1/n, 1].">A176502</a> * <a href="/A176504" title="a(n) = m + k where prime(m)*prime(k) = semiprime(n).">A176504</a> <a href="/A176505" title="Triangle T(n,m) read by rows: T(n,m) = a(n) - a(m) - a(n-m) + 1, where a(n) = A000931(n+4).">A176505</a> <a href="/A176506" title="Difference between the prime indices of the two factors of the n-th semiprime.">A176506</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:N._J._A._Sloane">N. J. A. Sloane</a>, Dec 07 2010</div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Added terms beyond a(13)=848, <a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, Dec 18 2012</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 April 6 16:52 EDT 2025. Contains 382544 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>