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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> </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> 57 </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 Ca莽茫o, Helmuth R. Malonek, Maria Irene Falc茫o, and Gra莽a 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 April 10 16:35 EDT 2025. Contains 382673 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>