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A047202 - 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>A047202 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A047202" 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=%2fA047202">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="A047202 - OEIS"></a> </div> <div class="motdbox"> <div class="motd"> <p>Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).</p> </div> <div class="donate"> <div id="donate-button-container"> <div id="donate-button"></div> <script src="https://www.paypalobjects.com/donate/sdk/donate-sdk.js" charset="UTF-8"></script> <script> PayPal.Donation.Button({ env:'production', hosted_button_id:'SVPGSDDCJ734A', image: { src:'https://www.paypalobjects.com/en_US/i/btn/btn_donateCC_LG.gif', alt:'Donate with PayPal button', title:'PayPal - The safer, easier way to pay online!', } }).render('#donate-button'); </script> </div> <a href="https://oeisf.org/donate/"> <strong>Other ways to Give</strong> </a> </div> </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> A047202 </div> <div class=seqname> Numbers that are congruent to {2, 3, 4} mod 5. </div> </div> <div class=scorerefs> 34 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>2, 3, 4, 7, 8, 9, 12, 13, 14, 17, 18, 19, 22, 23, 24, 27, 28, 29, 32, 33, 34, 37, 38, 39, 42, 43, 44, 47, 48, 49, 52, 53, 54, 57, 58, 59, 62, 63, 64, 67, 68, 69, 72, 73, 74, 77, 78, 79, 82, 83, 84, 87, 88, 89, 92, 93, 94, 97, 98, 99, 102, 103, 104, 107, 108</div> <div class=seqdatalinks> (<a href="/A047202/list">list</a>; <a href="/A047202/graph">graph</a>; <a href="/search?q=A047202+-id:A047202">refs</a>; <a href="/A047202/listen">listen</a>; <a href="/history?seq=A047202">history</a>; <a href="/search?q=id:A047202&fmt=text">text</a>; <a href="/A047202/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,1</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Stefano Spezia, <a href="/A047202/b047202.txt">Table of n, a(n) for n = 1..10000</a></div> <div class=sectline><a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>G.f.: x*(2+x+x^2+x^3) / ((1+x+x^2)*(x-1)^2). - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Oct 07 2011</div> <div class=sectline>From <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Jun 14 2016: (Start)</div> <div class=sectline>a(n) = a(n-1) + a(n-3) - a(n-4) for n>3.</div> <div class=sectline>a(n) = (15*n-3-6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9.</div> <div class=sectline>a(3k) = 5k-1, a(3k-1) = 5k-2, a(3k-2) = 5k-3. (End)</div> <div class=sectline>a(n) = 2*n - floor((n-1)/3) - ((n-1) mod 3). - <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Sep 26 2017</div> <div class=sectline>Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt((5+sqrt(5))/10)*Pi/5 + log(phi)/sqrt(5) - 3*log(2)/5, where phi is the golden ratio (<a href="/A001622" title="Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2.">A001622</a>). - <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Apr 16 2023</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline><a href="/A047202" title="Numbers that are congruent to {2, 3, 4} mod 5.">A047202</a>:=n->(15*n-3-6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9: seq(<a href="/A047202" title="Numbers that are congruent to {2, 3, 4} mod 5.">A047202</a>(n), n=1..100); # <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Jun 14 2016</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Select[Range[0, 200], MemberQ[{2, 3, 4}, Mod[#, 5]] &] (* <a href="/wiki/User:Vladimir_Joseph_Stephan_Orlovsky">Vladimir Joseph Stephan Orlovsky</a>, Feb 12 2012 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Magma) [n: n in [1..150] | n mod 5 in [2..4]]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Mar 31 2011</div> <div class=sectline>(PARI) a(n)=n\3*5+[-1, 2, 3][n%3+1] \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Dec 22 2011</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A001622" title="Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2.">A001622</a>.</div> <div class=sectline>Sequence in context: <a href="/A061856" title="The positions of the terms of A061855 in the sequence A014486, terms fixed by the permutation A057164.">A061856</a> <a href="/A105941" title="Powers of Lucas numbers.">A105941</a> <a href="/A276876" title="Sums-complement of the Beatty sequence for 2e.">A276876</a> * <a href="/A064953" title="Where the even terms appear in A064413.">A064953</a> <a href="/A097503" title="Numbers k such that A000203(k) = sigma(k) is not divisible by 6.">A097503</a> <a href="/A030701" title="Decimal expansion of 4^n contains no zeros (probably finite).">A030701</a></div> <div class=sectline>Adjacent sequences: <a href="/A047199" title="Number of nonempty subsets of {1,2,...,n} in which exactly 2/5 of the elements are <= n/3.">A047199</a> <a href="/A047200" title="Number of nonempty subsets of {1,2,...,n} in which exactly 3/5 of the elements are <= n/3.">A047200</a> <a href="/A047201" title="Numbers not divisible by 5.">A047201</a> * <a href="/A047203" title="Numbers that are congruent to {0, 2, 3, 4} mod 5.">A047203</a> <a href="/A047204" title="Numbers that are congruent to {3, 4} mod 5.">A047204</a> <a href="/A047205" title="Numbers that are congruent to {0, 3, 4} mod 5.">A047205</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>,<span title="it is very easy to produce terms of sequence">easy</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></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 December 3 18:04 EST 2024. Contains 378391 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>