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A005320 - 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>A005320 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A005320" 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=%2fA005320">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="A005320 - 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> A005320 </div> <div class=seqname> a(n) = 4*a(n-1) - a(n-2), with a(0) = 0, a(1) = 3. <br><font size=-1>(Formerly M2919)</font> </div> </div> <div class=scorerefs> 14 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 3, 12, 45, 168, 627, 2340, 8733, 32592, 121635, 453948, 1694157, 6322680, 23596563, 88063572, 328657725, 1226567328, 4577611587, 17083879020, 63757904493, 237947738952, 888033051315, 3314184466308, 12368704813917, 46160634789360, 172273834343523, 642934702584732, 2399464975995405, 8954925201396888, 33420235829592147</div> <div class=seqdatalinks> (<a href="/A005320/list">list</a>; <a href="/A005320/graph">graph</a>; <a href="/search?q=A005320+-id:A005320">refs</a>; <a href="/A005320/listen">listen</a>; <a href="/history?seq=A005320">history</a>; <a href="/search?q=id:A005320&fmt=text">text</a>; <a href="/A005320/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>For n > 1, a(n-1) is the determinant of the n X n band matrix which has {2,4,4,...,4,4,2} on the diagonal and a 1 on the entire super- and subdiagonal. This matrix appears when constructing a natural cubic spline interpolating n equally spaced data points. - g.degroot(AT)phys.uu.nl, Feb 14 2007</div> <div class=sectline>Integer values of x that make 9+3*x^2 a perfect square. - <a href="/wiki/User:Lorenz_H._Menke,_Jr.">Lorenz H. Menke, Jr.</a>, Mar 26 2008</div> <div class=sectline>The intermediate convergents to 3^(1/2), beginning with 3/2, 12/7, 45/26, 168/97, comprise a strictly increasing sequence whose numerators are the terms of this sequence and denominators are <a href="/A001075" title="a(0) = 1, a(1) = 2, a(n) = 4*a(n-1) - a(n-2).">A001075</a>. - <a href="/wiki/User:Clark_Kimberling">Clark Kimberling</a>, Aug 27 2008</div> <div class=sectline>a(n) also give the altitude to the middle side of a Super-Heronian Triangle. - <a href="/wiki/User:Johannes_Boot">Johannes Boot</a>, Oct 14 2010</div> <div class=sectline>a(n) gives values of y satisfying 3*x^2 - 4*y^2 = 12; corresponding x values are given by <a href="/A003500" title="a(n) = 4*a(n-1) - a(n-2) with a(0) = 2, a(1) = 4.">A003500</a>. - <a href="/wiki/User:Sture_Sj枚stedt">Sture Sj枚stedt</a>, Dec 19 2017</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Vincenzo Librandi, <a href="/A005320/b005320.txt">Table of n, a(n) for n = 0..1000</a></div> <div class=sectline>C. Banderier and D. Merlini, <a href="http://algo.inria.fr/banderier/Papers/infjumps.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, Melbourne, 2002.</div> <div class=sectline>Hac猫ne Belbachir, Soumeya Merwa Tebtoub, L谩szl贸 N茅meth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.</div> <div class=sectline>I. M. Gessel, Ji Li, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Gessel/gessel6.html">Compositions and Fibonacci identities</a>, J. Int. Seq. 16 (2013) 13.4.5</div> <div class=sectline>Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a></div> <div class=sectline>Clark Kimberling, <a href="http://dx.doi.org/10.1007/s000170050020">Best lower and upper approximates to irrational numbers</a>, Elemente der Mathematik, 52 (1997) 122-126.</div> <div class=sectline>E. Keith Lloyd, <a href="http://www.jstor.org/stable/3619201">The Standard Deviation of 1, 2,..., n: Pell's Equation and Rational Triangles</a>, Math. Gaz. vol 81 (1997), 231-243.</div> <div class=sectline>Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de s茅ries g茅n茅ratrices et quelques conjectures</a>, Dissertation, Universit茅 du Qu茅bec 脿 Montr茅al, 1992; arXiv:0911.4975 [math.NT], 2009.</div> <div class=sectline>Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992</div> <div class=sectline>William H. Richardson, <a href="http://www.math.wichita.edu/~richardson/heronian/heronian.html"> Super-Heronian Triangles</a> from Johannes Boot, Oct 14 2010</div> <div class=sectline><a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = (sqrt(3)/2)*( (2+sqrt(3))^n - (2-sqrt(3))^n ). - <a href="/wiki/User:Antonio_Alberto_Olivares">Antonio Alberto Olivares</a>, Jan 17 2004</div> <div class=sectline>G.f.: 3*x/(1-4*x+x^2). - <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Mar 04 2012</div> <div class=sectline>a(n) = 3*<a href="/A001353" title="a(n) = 4*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.">A001353</a>(n). - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Mar 14 2016</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline><a href="/A005320" title="a(n) = 4*a(n-1) - a(n-2), with a(0) = 0, a(1) = 3.">A005320</a>:=3*z/(1-4*z+z**2); # <a href="/wiki/User:Simon_Plouffe">Simon Plouffe</a> in his 1992 dissertation</div> <div class=sectline>a:= n-> (Matrix([[3, 0]]). Matrix([[4, 1], [ -1, 0]])^n)[1, 2]: seq(a(n), n=0..50); # <a href="/wiki/User:Alois_P._Heinz">Alois P. Heinz</a>, Aug 14 2008</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>LinearRecurrence[{4, -1}, {0, 3}, 40] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Mar 04 2012 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) Vec(3/(x^2-4*x+1)+O(x^99)) \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Mar 05 2012</div> <div class=sectline>(Magma) [3*Evaluate(ChebyshevSecond(n), 2): n in [0..40]]; // <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Oct 10 2022</div> <div class=sectline>(SageMath) [3*chebyshev_U(n-1, 2) for n in range(41)] # <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Oct 10 2022</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A001075" title="a(0) = 1, a(1) = 2, a(n) = 4*a(n-1) - a(n-2).">A001075</a>, <a href="/A001353" title="a(n) = 4*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.">A001353</a>, <a href="/A002194" title="Decimal expansion of sqrt(3).">A002194</a>, <a href="/A003500" title="a(n) = 4*a(n-1) - a(n-2) with a(0) = 2, a(1) = 4.">A003500</a>, <a href="/A082841" title="a(n) = 4*a(n-1) - a(n-2) for n>1, a(0)=3, a(1)=9.">A082841</a>.</div> <div class=sectline>Sequence in context: <a href="/A258626" title="Number of length n 1..(3+1) arrays with every leading partial sum divisible by 2, 3, 5 or 7">A258626</a> <a href="/A380047" title="E.g.f. A(x) satisfies A(x) = 1 + 3*x*exp(x)*A(x)^(1/3).">A380047</a> <a href="/A064017" title="Number of ternary trees (A001764) with n nodes and maximal diameter.">A064017</a> * <a href="/A062561" title="a(n) = 3*binomial(2*n, n-1).">A062561</a> <a href="/A128593" title="Column 1 of triangle A128592; a(n) = coefficient of q^(n+2) in the q-analog of the odd double factorials (2n+3)!! for n>=0.">A128593</a> <a href="/A359813" title="Number of primes < 10^n with exactly one odd decimal digit.">A359813</a></div> <div class=sectline>Adjacent sequences: <a href="/A005317" title="a(n) = (2^n + C(2*n,n))/2.">A005317</a> <a href="/A005318" title="Conway-Guy sequence: a(n + 1) = 2a(n) - a(n - floor( 1/2 + sqrt(2n) )).">A005318</a> <a href="/A005319" title="a(n) = 6*a(n-1) - a(n-2).">A005319</a> * <a href="/A005321" title="Upper triangular n X n (0,1)-matrices with no zero rows or columns.">A005321</a> <a href="/A005322" title="Column of Motzkin triangle.">A005322</a> <a href="/A005323" title="Column of Motzkin triangle.">A005323</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>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Typo in definition corrected by <a href="/wiki/User:Johannes_Boot">Johannes Boot</a>, Feb 05 2009</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 March 13 19:40 EDT 2025. 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