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A003500 - 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>A003500 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A003500" 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=%2fA003500">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="A003500 - 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> A003500 </div> <div class=seqname> a(n) = 4*a(n-1) - a(n-2) with a(0) = 2, a(1) = 4. <br><font size=-1>(Formerly M1278)</font> </div> </div> <div class=scorerefs> 51 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>2, 4, 14, 52, 194, 724, 2702, 10084, 37634, 140452, 524174, 1956244, 7300802, 27246964, 101687054, 379501252, 1416317954, 5285770564, 19726764302, 73621286644, 274758382274, 1025412242452, 3826890587534, 14282150107684, 53301709843202, 198924689265124</div> <div class=seqdatalinks> (<a href="/A003500/list">list</a>; <a href="/A003500/graph">graph</a>; <a href="/search?q=A003500+-id:A003500">refs</a>; <a href="/A003500/listen">listen</a>; <a href="/history?seq=A003500">history</a>; <a href="/search?q=id:A003500&fmt=text">text</a>; <a href="/A003500/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,1</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>a(n) gives values of x satisfying x^2 - 3*y^2 = 4; corresponding y values are given by 2*<a href="/A001353" title="a(n) = 4*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.">A001353</a>(n).</div> <div class=sectline>If M is any given term of the sequence, then the next one is 2*M + sqrt(3*M^2 - 12). - <a href="/wiki/User:Lekraj_Beedassy">Lekraj Beedassy</a>, Feb 18 2002</div> <div class=sectline>For n > 0, the three numbers a(n) - 1, a(n), and a(n) + 1 form a Fleenor-Heronian triangle, i.e., a Heronian triangle with consecutive sides, whose area A(n) may be obtained from the relation [4*A(n)]^2 = 3([a(2n)]^2 - 4); or 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>(2*n)/2 and whose semiperimeter is 3*a[n]/2. The sequence is symmetrical about a[0], i.e., a[-n] = a[n].</div> <div class=sectline>For n > 0, a(n) + 2 is the number of dimer tilings of a 2*n X 2 Klein bottle (cf. <a href="/A103999" title="Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2M x 2N Klein bottle.">A103999</a>).</div> <div class=sectline>Tsumura shows that, for prime p, a(p) is composite (contrary to a conjecture of Juricevic). - <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Apr 13 2010</div> <div class=sectline>Except for the first term, positive values of x (or y) satisfying x^2 - 4*x*y + y^2 + 12 = 0. - <a href="/wiki/User:Colin_Barker">Colin Barker</a>, Feb 04 2014</div> <div class=sectline>Except for the first term, positive values of x (or y) satisfying x^2 - 14*x*y + y^2 + 192 = 0. - <a href="/wiki/User:Colin_Barker">Colin Barker</a>, Feb 16 2014</div> <div class=sectline><a href="/A268281" title="Numbers n such that n-tau(n), phi(n) and n form a Heronian triangle, where tau=A000005 is the number of divisors and phi=A00...">A268281</a>(n) - 1 is a member of this sequence iff <a href="/A268281" title="Numbers n such that n-tau(n), phi(n) and n form a Heronian triangle, where tau=A000005 is the number of divisors and phi=A00...">A268281</a>(n) is prime. - <a href="/wiki/User:Frank_M_Jackson">Frank M Jackson</a>, Feb 27 2016</div> <div class=sectline>a(n) gives values of x satisfying 3*x^2 - 4*y^2 = 12; corresponding y values are given by <a href="/A005320" title="a(n) = 4*a(n-1) - a(n-2), with a(0) = 0, a(1) = 3.">A005320</a>. - <a href="/wiki/User:Sture_Sj枚stedt">Sture Sj枚stedt</a>, Dec 19 2017</div> <div class=sectline>Middle side lengths of almost-equilateral Heronian triangles. - <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, May 20 2020</div> <div class=sectline>For all elements k of the sequence, 3*(k-2)*(k+2) is a square. - <a href="/wiki/User:Davide_Rotondo">Davide Rotondo</a>, Oct 25 2020</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 82.</div> <div class=sectline>J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p.91.</div> <div class=sectline>Michael P. Cohen, Generating Heronian Triangles With Consecutive Integer Sides. Journal of Recreational Mathematics, vol. 30 no. 2 1999-2000 p. 123.</div> <div class=sectline>L. E. Dickson, History of The Theory of Numbers, Vol. 2 pp. 197;198;200;201. Chelsea NY.</div> <div class=sectline>Charles R. Fleenor, Heronian Triangles with Consecutive Integer Sides, Journal of Recreational Mathematics, Volume 28, no. 2 (1996-7) 113-115.</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> <div class=sectline>R. P. Stanley. Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999.</div> <div class=sectline>V. D. To, "Finding All Fleenor-Heronian Triangles", Journal of Recreational Mathematics vol. 32 no.4 2003-4 pp. 298-301 Baywood NY.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>T. D. Noe, <a href="/A003500/b003500.txt">Table of n, a(n) for n=0..200</a></div> <div class=sectline>P. Bala, <a href="/A174500/a174500_2.pdf">Some simple continued fraction expansions for an infinite product, Part 1</a></div> <div class=sectline>R. A. Beauregard and E. R. Suryanarayan, <a href="http://www.maa.org/sites/default/files/pdf/mathdl/CMJ/methodoflastresort.pdf">The Brahmagupta Triangles</a>, The College Mathematics Journal 29(1) 13-7 1998 MAA.</div> <div class=sectline>Hac猫ne Belbachir, Soumeya Merwa Tebtoub and 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>Daniel Birmajer, Juan B. Gil and Michael D. Weiner, <a href="http://arxiv.org/abs/1505.06339">Linear recurrence sequences with indices in arithmetic progression and their sums</a>, arXiv preprint arXiv:1505.06339 [math.NT], 2015.</div> <div class=sectline>K. S. Brown's Mathpages, <a href="http://www.mathpages.com/home/kmath480/kmath480.htm">Some Properties of the Lucas Sequence(2, 4, 14, 52, 194, ...)</a></div> <div class=sectline>H. W. Gould, <a href="http://www.fq.math.ca/Scanned/11-1/gould.pdf">A triangle with integral sides and area</a>, Fib. Quart., 11 (1973), 27-39.</div> <div class=sectline>Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a></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>S. Northshield, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Northshield/north4.html">An Analogue of Stern's Sequence for Z[sqrt(2)]</a>, Journal of Integer Sequences, 18 (2015), #15.11.6.</div> <div class=sectline>Hideyuki Ohtskua, proposer, <a href="https://www.fq.math.ca/Problems/FQElemProbAug2024.pdf">Problem B-1351</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 62, No. 3 (2024), p. 258.</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>Jeffrey Shallit, <a href="http://www.jstor.org/stable/2690344">An interesting continued fraction</a>, Math. Mag., 48 (1975), 207-211.</div> <div class=sectline>Jeffrey Shallit, <a href="/A005248/a005248_1.pdf">An interesting continued fraction</a>, Math. Mag., 48 (1975), 207-211. [Annotated scanned copy]</div> <div class=sectline>Yu Tsumura, <a href="http://arxiv.org/abs/1004.1244">On compositeness of special types of integers</a>, arXiv:1004.1244 [math.NT], 2010.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a></div> <div class=sectline>Wikipedia, <a href="https://en.wikipedia.org/wiki/Heronian_triangle">Heronian triangle</a></div> <div class=sectline>A. V. Zarelua, <a href="https://doi.org/10.1007/s11006-006-0090-y">On Matrix Analogs of Fermat's Little Theorem</a>, Mathematical Notes, vol. 79, no. 6, 2006, pp. 783-796. Translated from Matematicheskie Zametki, vol. 79, no. 6, 2006, pp. 840-855.</div> <div class=sectline><a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a></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) = ( 2 + sqrt(3) )^n + ( 2 - sqrt(3) )^n.</div> <div class=sectline>a(n) = 2*<a href="/A001075" title="a(0) = 1, a(1) = 2, a(n) = 4*a(n-1) - a(n-2).">A001075</a>(n).</div> <div class=sectline>G.f.: 2*(1 - 2*x)/(1 - 4*x + x^2). <a href="/wiki/User:Simon_Plouffe">Simon Plouffe</a> in his 1992 dissertation.</div> <div class=sectline>a(n) = <a href="/A001835" title="a(n) = 4*a(n-1) - a(n-2), with a(0) = 1, a(1) = 1.">A001835</a>(n) + <a href="/A001835" title="a(n) = 4*a(n-1) - a(n-2), with a(0) = 1, a(1) = 1.">A001835</a>(n+1).</div> <div class=sectline>a(n) = trace of n-th power of the 2 X 2 matrix [1 2 / 1 3]. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Jun 30 2003 [corrected by <a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, Jun 18 2020]</div> <div class=sectline>From the addition formula, a(n+m) = a(n)*a(m) - a(m-n), it is easy to derive multiplication formulas, such as: a(2*n) = (a(n))^2 - 2, a(3*n) = (a(n))^3 - 3*(a(n)), a(4*n) = (a(n))^4 - 4*(a(n))^2 + 2, a(5*n) = (a(n))^5 - 5*(a(n))^3 + 5*(a(n)), a(6*n) = (a(n))^6 - 6*(a(n))^4 + 9*(a(n))^2 - 2, etc. The absolute values of the coefficients in the expansions are given by the triangle <a href="/A034807" title="Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.">A034807</a>. - <a href="/wiki/User:John_Blythe_Dobson">John Blythe Dobson</a>, Nov 04 2007</div> <div class=sectline>a(n) = 2*<a href="/A001353" title="a(n) = 4*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.">A001353</a>(n+1) - 4*<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>, Nov 16 2007</div> <div class=sectline>From <a href="/wiki/User:Peter_Bala">Peter Bala</a>, Jan 06 2013: (Start)</div> <div class=sectline>Let F(x) = Product_{n=0..infinity} (1 + x^(4*n + 1))/(1 + x^(4*n + 3)). Let alpha = 2 - sqrt(3). This sequence gives the simple continued fraction expansion of 1 + F(alpha) = 2.24561 99455 06551 88869 ... = 2 + 1/(4 + 1/(14 + 1/(52 + ...))). Cf. <a href="/A174500" title="Continued fraction expansion for exp( Sum_{n>=1} 1/(n*A003500(n)) ), where A003500(n) = (2+sqrt(3))^n + (2-sqrt(3))^n.">A174500</a>.</div> <div class=sectline>Also F(-alpha) = 0.74544 81786 39692 68884 ... has the continued fraction representation 1 - 1/(4 - 1/(14 - 1/(52 - ...))) and the simple continued fraction expansion 1/(1 + 1/((4 - 2) + 1/(1 + 1/((14 - 2) + 1/(1 + 1/((52 - 2) + 1/(1 + ...))))))).</div> <div class=sectline>F(alpha)*F(-alpha) has the simple continued fraction expansion 1/(1 + 1/((4^2 - 4) + 1/(1 + 1/((14^2 - 4) + 1/(1 + 1/((52^2 - 4) + 1/(1 + ...))))))).</div> <div class=sectline>(End)</div> <div class=sectline>a(2^n) = <a href="/A003010" title="A Lucas-Lehmer sequence: a(0) = 4; for n>0, a(n) = a(n-1)^2 - 2.">A003010</a>(n). - <a href="/wiki/User:John_Blythe_Dobson">John Blythe Dobson</a>, Mar 10 2014</div> <div class=sectline>a(n) = [x^n] ( (1 + 4*x + sqrt(1 + 8*x + 12*x^2))/2 )^n for n >= 1. - <a href="/wiki/User:Peter_Bala">Peter Bala</a>, Jun 23 2015</div> <div class=sectline>E.g.f.: 2*exp(2*x)*cosh(sqrt(3)*x). - <a href="/wiki/User:Ilya_Gutkovskiy">Ilya Gutkovskiy</a>, Apr 27 2016</div> <div class=sectline>a(n) = Sum_{k=0..floor(n/2)} (-1)^k*n*(n - k - 1)!/(k!*(n - 2*k)!)*4^(n - 2*k) for n >= 1. - <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, May 10 2016</div> <div class=sectline>From <a href="/wiki/User:Peter_Bala">Peter Bala</a>, Oct 15 2019: (Start)</div> <div class=sectline>a(n) = trace(M^n), where M is the 2 X 2 matrix [0, 1; -1, 4].</div> <div class=sectline>Consequently the Gauss congruences hold: a(n*p^k) = a(n*p^(k-1)) ( mod p^k ) for all prime p and positive integers n and k. See Zarelua and also Stanley (Ch. 5, Ex. 5.2(a) and its solution).</div> <div class=sectline>2*Sum_{n >= 1} 1/( a(n) - 6/a(n) ) = 1.</div> <div class=sectline>6*Sum_{n >= 1} (-1)^(n+1)/( a(n) + 2/a(n) ) = 1.</div> <div class=sectline>8*Sum_{n >= 1} 1/( a(n) + 24/(a(n) - 12/(a(n))) ) = 1.</div> <div class=sectline>8*Sum_{n >= 1} (-1)^(n+1)/( a(n) + 8/(a(n) + 4/(a(n))) ) = 1.</div> <div class=sectline>Series acceleration formulas for sums of reciprocals:</div> <div class=sectline>Sum_{n >= 1} 1/a(n) = 1/2 - 6*Sum_{n >= 1} 1/(a(n)*(a(n)^2 - 6)),</div> <div class=sectline>Sum_{n >= 1} 1/a(n) = 1/8 + 24*Sum_{n >= 1} 1/(a(n)*(a(n)^2 + 12)),</div> <div class=sectline>Sum_{n >= 1} (-1)^(n+1)/a(n) = 1/6 + 2*Sum_{n >= 1} (-1)^(n+1)/(a(n)*(a(n)^2 + 2)) and</div> <div class=sectline>Sum_{n >= 1} (-1)^(n+1)/a(n) = 1/8 + 8*Sum_{n >= 1} (-1)^(n+1)/(a(n)*(a(n)^2 + 12)).</div> <div class=sectline>Sum_{n >= 1} 1/a(n) = ( theta_3(2-sqrt(3))^2 - 1 )/4 = 0.34770 07561 66992 06261 .... See Borwein and Borwein, Proposition 3.5 (i), p.91.</div> <div class=sectline>Sum_{n >= 1} (-1)^(n+1)/a(n) = ( 1 - theta_3(sqrt(3)-2)^2 )/4. Cf. <a href="/A003499" title="a(n) = 6*a(n-1) - a(n-2), with a(0) = 2, a(1) = 6.">A003499</a> and <a href="/A153415" title="Decimal expansion of Sum_{n>=1} 1/A000032(2*n).">A153415</a>. (End)</div> <div class=sectline>a(n) = tan(Pi/12)^n + tan(5*Pi/12)^n. - <a href="/wiki/User:Greg_Dresden">Greg Dresden</a>, Oct 01 2020</div> <div class=sectline>From <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Sep 06 2021: (Start)</div> <div class=sectline>a(n) = S(n, 4) - S(n-2, 4) = 2*T(n, 2), for n >= 0, with S and T Chebyshev polynomials, with S(-1, x) = 0 and S(-2, x) = -1. S(n, 4) = <a href="/A001353" title="a(n) = 4*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.">A001353</a>(n+1), for n >= -1, and T(n, 2) = <a href="/A001075" title="a(0) = 1, a(1) = 2, a(n) = 4*a(n-1) - a(n-2).">A001075</a>(n).</div> <div class=sectline>a(2*k) = <a href="/A067902" title="a(n) = 14*a(n-1) - a(n-2); a(0) = 2, a(1) = 14.">A067902</a>(k), a(2*k+1) = 4*<a href="/A001570" title="Numbers k such that k^2 is centered hexagonal.">A001570</a>(k+1), for k >= 0. (End)</div> <div class=sectline>a(n) = sqrt(2 + 2*<a href="/A011943" title="Numbers k such that any group of k consecutive integers has integral standard deviation (viz. A011944(k)).">A011943</a>(n+1)) = sqrt(2 + 2*<a href="/A102344" title="Numbers n such that the Diophantine equation (x+2)^3-x^3=2*n^2 has solutions.">A102344</a>(n+1)), n>0. - <a href="/wiki/User:Ralf_Steiner">Ralf Steiner</a>, Sep 23 2021</div> <div class=sectline>Sum_{n>=1} arctan(3/a(n)^2) = Pi/6 - arctan(1/3) = <a href="/A019673" title="Decimal expansion of Pi/6.">A019673</a> - <a href="/A105531" title="Decimal expansion of arctan 1/3.">A105531</a> (Ohtskua, 2024). - <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Aug 29 2024</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline><a href="/A003500" title="a(n) = 4*a(n-1) - a(n-2) with a(0) = 2, a(1) = 4.">A003500</a> := proc(n) option remember; if n <= 1 then 2*n+2 else 4*procname(n-1)-procname(n-2); fi;</div> <div class=sectline>end proc;</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>a[0]=2; a[1]=4; a[n_]:= a[n]= 4a[n-1] -a[n-2]; Table[a[n], {n, 0, 23}]</div> <div class=sectline>LinearRecurrence[{4, -1}, {2, 4}, 30] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Aug 20 2011 *)</div> <div class=sectline>Table[Round@LucasL[2n, Sqrt[2]], {n, 0, 20}] (* <a href="/wiki/User:Vladimir_Reshetnikov">Vladimir Reshetnikov</a>, Sep 15 2016 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Sage) [lucas_number2(n, 4, 1) for n in range(0, 24)] # <a href="/wiki/User:Zerinvary_Lajos">Zerinvary Lajos</a>, May 14 2009</div> <div class=sectline>(Haskell)</div> <div class=sectline>a003500 n = a003500_list !! n</div> <div class=sectline>a003500_list = 2 : 4 : zipWith (-)</div> <div class=sectline> (map (* 4) $ tail a003500_list) a003500_list</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Dec 17 2011</div> <div class=sectline>(PARI) x='x+O('x^99); Vec(-2*(-1+2*x)/(1-4*x+x^2)) \\ <a href="/wiki/User:Altug_Alkan">Altug Alkan</a>, Apr 04 2016</div> <div class=sectline>(Magma) I:=[2, 4]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in [1..30]]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Nov 14 2018</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="/A001835" title="a(n) = 4*a(n-1) - a(n-2), with a(0) = 1, a(1) = 1.">A001835</a>.</div> <div class=sectline>Cf. <a href="/A011945" title="Areas of almost-equilateral Heronian triangles (integral side lengths m-1, m, m+1 and integral area).">A011945</a> (areas), <a href="/A334277" title="Perimeters of almost-equilateral Heronian triangles.">A334277</a> (perimeters).</div> <div class=sectline>Cf. this sequence (middle side lengths), <a href="/A016064" title="Smallest side lengths of almost-equilateral Heronian triangles (sides are consecutive positive integers, area is a nonnegati...">A016064</a> (smallest side lengths), <a href="/A335025" title="Largest side lengths of almost-equilateral Heronian triangles.">A335025</a> (largest side lengths).</div> <div class=sectline>Cf. <a href="/A001570" title="Numbers k such that k^2 is centered hexagonal.">A001570</a>, <a href="/A002530" title="a(n) = 4*a(n-2) - a(n-4) for n > 1, a(n) = n for n = 0, 1.">A002530</a>, <a href="/A005320" title="a(n) = 4*a(n-1) - a(n-2), with a(0) = 0, a(1) = 3.">A005320</a>, <a href="/A006051" title="Square hex numbers.">A006051</a>, <a href="/A048788" title="a(2n+1) = a(2n) + a(2n-1), a(2n) = 2*a(2n-1) + a(2n-2); a(n) = n for n = 0, 1.">A048788</a>, <a href="/A174500" title="Continued fraction expansion for exp( Sum_{n>=1} 1/(n*A003500(n)) ), where A003500(n) = (2+sqrt(3))^n + (2-sqrt(3))^n.">A174500</a>, <a href="/A268281" title="Numbers n such that n-tau(n), phi(n) and n form a Heronian triangle, where tau=A000005 is the number of divisors and phi=A00...">A268281</a>.</div> <div class=sectline>Cf. <a href="/A011943" title="Numbers k such that any group of k consecutive integers has integral standard deviation (viz. A011944(k)).">A011943</a>, <a href="/A102344" title="Numbers n such that the Diophantine equation (x+2)^3-x^3=2*n^2 has solutions.">A102344</a>, <a href="/A019673" title="Decimal expansion of Pi/6.">A019673</a>, <a href="/A105531" title="Decimal expansion of arctan 1/3.">A105531</a>.</div> <div class=sectline>Sequence in context: <a href="/A046650" title="Number of rooted planar maps.">A046650</a> <a href="/A327235" title="Number of unlabeled simple graphs with n vertices whose edge-set is not connected.">A327235</a> <a href="/A055727" title="Number of lucky 4,6 triples <= 10^n.">A055727</a> * <a href="/A316363" title="O.g.f. A(x) satisfies: Sum_{n>=1} (x + (-1)^n*A(x))^n / n = 0.">A316363</a> <a href="/A295760" title="G.f. A(x) satisfies: A(x - A(x^2)) = x + A(x^2).">A295760</a> <a href="/A129876" title="Sequence i_n arising in enumeration of arrays of directed blocks (see Quaintance reference for precise definition).">A129876</a></div> <div class=sectline>Adjacent sequences: <a href="/A003497" title="Internal energy series for b.c.c. lattice.">A003497</a> <a href="/A003498" title="High temperature series for internal energy for spherical model on f.c.c. lattice.">A003498</a> <a href="/A003499" title="a(n) = 6*a(n-1) - a(n-2), with a(0) = 2, a(1) = 6.">A003499</a> * <a href="/A003501" title="a(n) = 5*a(n-1) - a(n-2), with a(0) = 2, a(1) = 5.">A003501</a> <a href="/A003502" title="The smaller of a betrothed pair.">A003502</a> <a href="/A003503" title="The larger of a betrothed pair.">A003503</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>,<span title="an exceptionally nice sequence">nice</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>More terms from <a href="/wiki/User:James_A._Sellers">James A. Sellers</a>, May 03 2000</div> <div class=sectline>Additional comments from <a href="/wiki/User:Lekraj_Beedassy">Lekraj Beedassy</a>, Feb 14 2002</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 November 24 17:37 EST 2024. Contains 378083 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>