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A000466 - 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>A000466 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A000466" 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=%2fA000466">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="A000466 - 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> A000466 </div> <div class=seqname> a(n) = 4*n^2 - 1. </div> </div> <div class=scorerefs> 54 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>-1, 3, 15, 35, 63, 99, 143, 195, 255, 323, 399, 483, 575, 675, 783, 899, 1023, 1155, 1295, 1443, 1599, 1763, 1935, 2115, 2303, 2499, 2703, 2915, 3135, 3363, 3599, 3843, 4095, 4355, 4623, 4899, 5183, 5475, 5775, 6083, 6399, 6723, 7055, 7395</div> <div class=seqdatalinks> (<a href="/A000466/list">list</a>; <a href="/A000466/graph">graph</a>; <a href="/search?q=A000466+-id:A000466">refs</a>; <a href="/A000466/listen">listen</a>; <a href="/history?seq=A000466">history</a>; <a href="/search?q=id:A000466&fmt=text">text</a>; <a href="/A000466/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,2</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Sum_{n>=1} (-1)^n*a(n)/n! = 1 - 1/e = <a href="/A068996" title="Decimal expansion of 1 - 1/e.">A068996</a>. - <a href="/wiki/User:Gerald_McGarvey">Gerald McGarvey</a>, Nov 06 2007</div> <div class=sectline>Sequence arises from reading the line from -1, in the direction -1, 15, ... and the same line from 3, in the direction 3, 35, ..., in the square spiral whose nonnegative vertices are the squares <a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, May 24 2008</div> <div class=sectline>a(n) is the product of the consecutive odd integers 2n-1 and 2n+1 (cf. <a href="/A005408" title="The odd numbers: a(n) = 2*n + 1.">A005408</a>). - <a href="/wiki/User:Doug_Bell">Doug Bell</a>, Mar 08 2009</div> <div class=sectline>For n>0: a(n) = <a href="/A176271" title="The odd numbers as a triangle read by rows.">A176271</a>(2*n,n); cf. <a href="/A016754" title="Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.">A016754</a>, <a href="/A053755" title="a(n) = 4*n^2 + 1.">A053755</a>. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Apr 13 2010</div> <div class=sectline>a(n+1) gives the curvature c(n) of the n-th circle touching the two equal semicircles of the symmetric arbelos (1/2, 1/2) and the (n-1)-st circle, with input c(0) = 3 = <a href="/A059100" title="a(n) = n^2 + 2.">A059100</a>(1) (referring to the second circle of the Pappus chain), for n >= 0. - <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a> and <a href="/wiki/User:Kival_Ngaokrajang">Kival Ngaokrajang</a>, Jul 03 2015</div> <div class=sectline>After 3, a(n) is pseudoprime to base 2n. For example: (2*2)^(a(2)-1) == 1 (mod a(2)), in fact 4^14 = 15*17895697+1. - <a href="/wiki/User:Bruno_Berselli">Bruno Berselli</a>, Sep 24 2015</div> <div class=sectline>Numbers m such that m+1 and (m+1)/4 are squares. - <a href="/wiki/User:Bruno_Berselli">Bruno Berselli</a>, Mar 03 2016</div> <div class=sectline>After -1, the least common multiple of 2*m+1 and 2*m-1. - <a href="/wiki/User:Colin_Barker">Colin Barker</a>, Feb 11 2017</div> <div class=sectline>This sequence contains all products of the twin prime pairs (see <a href="/A037074" title="Numbers that are the product of a pair of twin primes.">A037074</a>). - <a href="/wiki/User:Charles_Kusniec">Charles Kusniec</a>, Oct 03 2019</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.</div> <div class=sectline>L. B. W. Jolley, Summation of Series, Dover, 2nd ed., 1961.</div> <div class=sectline>Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), pp. 980-981.</div> <div class=sectline>A. Languasco and A. Zaccagnini, Manuale di Crittografia, Ulrico Hoepli Editore (2015), p. 259.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Vincenzo Librandi, <a href="/A000466/b000466.txt">Table of n, a(n) for n = 0..900</a></div> <div class=sectline>Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, <a href="https://www.emis.de/journals/JIS/VOL21/Falcao/falcao2.html">Combinatorial Identities Associated with a Multidimensional Polynomial Sequence</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.</div> <div class=sectline>Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.</div> <div class=sectline>Kival Ngaokrajang, <a href="/A000466/a000466.pdf">Illustration of the Pappus chain (downwards direction)</a>.</div> <div class=sectline><a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>O.g.f.: ( 1-6*x-3*x^2 ) / (x-1)^3 . - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Mar 24 2011</div> <div class=sectline>E.g.f.: (-1 + 4*x + 4*x^2)*exp(x). - <a href="/wiki/User:Ilya_Gutkovskiy">Ilya Gutkovskiy</a>, May 26 2016</div> <div class=sectline>Sum_{n>=1} 1/a(n) = 1/2 [Jolley eq. 233]. - <a href="/wiki/User:Benoit_Cloitre">Benoit Cloitre</a>, Apr 05 2002</div> <div class=sectline>Sum_{n>=1} 2/a(n) = 1 = 2/3 + 2/15 + 2/35 + 2/63 + 2/99 + 2/143, ..., with partial sums: 2/3, 4/5, 6/7, 8/9, 10/11, 12/13, 14/15, ... - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Jun 16 2003</div> <div class=sectline>1/3 + Sum_{n>=2} 4/a(n) = 1 = 1/3 + 4/15 + 4/35 + 4/63, ..., with partial sums: 1/3, 3/5, 5/7, 7/9, 9/11, ..., (2n+1)/(2n+3). - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Jun 18 2003</div> <div class=sectline>Sum_{n>=0} 2/a(2*n+1) = Pi/4 = 2/3 + 2/35 + 2/99, ... = (1 - 1/3) + (1/5 - 2/7) + (1/9 - 1/11) + ... = Sum_{n>=0} (-1)^n/(2*n+1). - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Jun 22 2003</div> <div class=sectline>Product(n>=1, (a(n)+1)/a(n)) = Pi/2 (Wallis formula). - Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Mar 03 2004</div> <div class=sectline>a(n)+2 = <a href="/A053755" title="a(n) = 4*n^2 + 1.">A053755</a>(n). - <a href="/wiki/User:Zak_Seidov">Zak Seidov</a>, Jan 16 2007</div> <div class=sectline>a(n)^2 + <a href="/A008586" title="Multiples of 4.">A008586</a>(n)^2 = <a href="/A053755" title="a(n) = 4*n^2 + 1.">A053755</a>(n)^2 (Pythagorean triple). - <a href="/wiki/User:Zak_Seidov">Zak Seidov</a>, Jan 16 2007</div> <div class=sectline>a(n) = a(n-1) + 8*n - 4 for n > 0, a(0)=-1. - <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Dec 17 2010</div> <div class=sectline>Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 - 1/2 = (<a href="/A019669" title="Decimal expansion of Pi/2.">A019669</a>-1)/2. [Jolley eq (366)]. - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Mar 24 2011</div> <div class=sectline>For n>0, a(n) = 2/(Integral_{x=0..Pi/2} (sin(x))^3*(cos(x))^(2*n-2)). - <a href="/wiki/User:Francesco_Daddi">Francesco Daddi</a>, Aug 02 2011</div> <div class=sectline>Nonlinear recurrence for c(n) = a(n+1) (see the arbelos comment above) from Descartes' three circle theorem (see the links under <a href="/A259555" title="a(n) = 2*n^2 - 2*n + 17.">A259555</a>): c(n) = 4 + c(n-1) + 4*sqrt(c(n-1) + 1), with input c(0) = 3 = <a href="/A059100" title="a(n) = n^2 + 2.">A059100</a>(1), for n >= 0. The appropriate solution of this recurrence is c(n-1) + 1 = 4*n^2. - <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Jul 03 2015</div> <div class=sectline>a(n) = 3*Pochhammer(5/2,n-1)/Pochhammer(1/2,n-1). Hence, the e.g.f. for a(n+1), i.e., dropping the first term, is 3* 1F1(5/2;1/2;x), with 1F1 being the confluent hypergeometric function (also known as Kummer's). - <a href="/wiki/User:Stanislav_Sykora">Stanislav Sykora</a>, May 26 2016</div> <div class=sectline>Product_{n>=1} (1 - 1/a(n)) = sin(Pi/sqrt(2))/sqrt(2). - <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Feb 04 2021</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline><a href="/A000466" title="a(n) = 4*n^2 - 1.">A000466</a>:=n->4*n^2-1; seq(<a href="/A000466" title="a(n) = 4*n^2 - 1.">A000466</a>(n), n=0..100); # <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Nov 19 2013</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>4 Range[0, 50]^2-1 (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Jan 23 2011 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Magma) [4*n^2-1: n in [0..50]]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Apr 26 2011</div> <div class=sectline>(PARI) a(n)=4*n^2-1 \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Oct 27 2011</div> <div class=sectline>(Maxima) makelist(4*n^2-1, n, 0, 50); /* <a href="/wiki/User:Martin_Ettl">Martin Ettl</a>, Nov 12 2012 */</div> <div class=sectline>(Sage) [4*n^2-1 for n in (0..50)] # <a href="/wiki/User:Bruno_Berselli">Bruno Berselli</a>, Sep 24 2015</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>, <a href="/A001539" title="a(n) = (4*n+1)*(4*n+3).">A001539</a>, <a href="/A016286" title="Inverse of 2277th cyclotomic polynomial.">A016286</a>, <a href="/A016742" title="Even squares: a(n) = (2*n)^2.">A016742</a>.</div> <div class=sectline>Factor of <a href="/A160466" title="Row sums of the Eta triangle A160464">A160466</a>. Superset of <a href="/A037074" title="Numbers that are the product of a pair of twin primes.">A037074</a>.</div> <div class=sectline>Cf. <a href="/A059100" title="a(n) = n^2 + 2.">A059100</a> (curvatures for a Pappus chain).</div> <div class=sectline>Sequence in context: <a href="/A317182" title="Consider integer triangles as listed in rows of table A316841. Sequence gives sorted and uniqued values of 16*area^2 for the...">A317182</a> <a href="/A236693" title="Numbers k such that 2^sigma(k) == 1 (mod k).">A236693</a> <a href="/A317183" title="Consider primitive integer triangles as listed in rows of table A316842. Sequence gives sorted and uniqued values of 16*area...">A317183</a> * <a href="/A241237" title="Number of isosceles triangles, distinct up to congruence, on a centered hexagonal grid of size n.">A241237</a> <a href="/A338351" title="Lexicographically earliest infinite sequence {a(n)} of distinct odd positive numbers such that, for n>2, a(n) has a common f...">A338351</a> <a href="/A145949" title="Number of n X n binary arrays symmetric under 90-degree rotation with all ones connected in a 3 X 2 elbow 1,1 1,2 1,3 2,3 in...">A145949</a></div> <div class=sectline>Adjacent sequences: <a href="/A000463" title="n followed by n^2.">A000463</a> <a href="/A000464" title="Expansion of e.g.f. sin(x)/cos(2*x).">A000464</a> <a href="/A000465" title="Number of bipartite partitions of n white objects and 4 black ones.">A000465</a> * <a href="/A000467" title="Number of permutations of [n] in which the longest increasing run has length 6.">A000467</a> <a href="/A000468" title="Powers of ten written in base 8.">A000468</a> <a href="/A000469" title="1 together with products of 2 or more distinct primes.">A000469</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="sequence contains negative numbers">sign</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>Chan Siu Kee (skchan5(AT)hkein.ie.cuhk.hk)</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 15:56 EST 2024. 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