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A011379 - 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>A011379 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A011379" 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=%2fA011379">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="A011379 - 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> A011379 </div> <div class=seqname> a(n) = n^2*(n+1). </div> </div> <div class=scorerefs> 61 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 2, 12, 36, 80, 150, 252, 392, 576, 810, 1100, 1452, 1872, 2366, 2940, 3600, 4352, 5202, 6156, 7220, 8400, 9702, 11132, 12696, 14400, 16250, 18252, 20412, 22736, 25230, 27900, 30752, 33792, 37026, 40460, 44100, 47952, 52022, 56316, 60840</div> <div class=seqdatalinks> (<a href="/A011379/list">list</a>; <a href="/A011379/graph">graph</a>; <a href="/search?q=A011379+-id:A011379">refs</a>; <a href="/A011379/listen">listen</a>; <a href="/history?seq=A011379">history</a>; <a href="/search?q=id:A011379&fmt=text">text</a>; <a href="/A011379/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>(1) a(n) = sum of second string of n triangular numbers - sum of first n triangular numbers, or the 2n-th partial sum of triangular numbers (<a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>) - the n-th partial sum of triangular numbers (<a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>). The same for natural numbers gives squares. (2) a(n) = (n-th triangular number)*(the n-th even number) = n(n+1)/2 * (2n). - <a href="/wiki/User:Amarnath_Murthy">Amarnath Murthy</a>, Nov 05 2002</div> <div class=sectline>Let M(n) be the n X n matrix m(i,j)=1/(i+j+x), let P(n,x) = (Product_{i=0..n-1} i!^2)/det(M(n)). Then P(n,x) is a polynomial with integer coefficients of degree n^2 and a(n) is the coefficient of x^(n^2-1). - <a href="/wiki/User:Benoit_Cloitre">Benoit Cloitre</a>, Jan 15 2003</div> <div class=sectline>Y values of solutions of the equation: (X-Y)^3-X*Y=0. X values are a(n)=n*(n+1)^2 (see <a href="/A045991" title="a(n) = n^3 - n^2.">A045991</a>) - <a href="/wiki/User:Mohamed_Bouhamida">Mohamed Bouhamida</a>, May 09 2006</div> <div class=sectline>a(2d-1) is the number of self-avoiding walk of length 3 in the d-dimensional hypercubic lattice. - <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Sep 06 2006</div> <div class=sectline>a(n) mod 10 is periodic 5: repeat [0, 2, 2, 6, 0]. - <a href="/wiki/User:Mohamed_Bouhamida">Mohamed Bouhamida</a>, Sep 05 2009</div> <div class=sectline>This sequence is related to <a href="/A005449" title="Second pentagonal numbers: a(n) = n*(3*n + 1)/2.">A005449</a> by a(n) = n*<a href="/A005449" title="Second pentagonal numbers: a(n) = n*(3*n + 1)/2.">A005449</a>(n)-sum(<a href="/A005449" title="Second pentagonal numbers: a(n) = n*(3*n + 1)/2.">A005449</a>(i), i=0..n-1), and this is the case d=3 in the identity n^2*(d*n+d-2)/2 - Sum_{k=0..n-1} k*(d*k+d-2)/2 = n*(n+d)*(2*d*n+d-3)/6. - <a href="/wiki/User:Bruno_Berselli">Bruno Berselli</a>, Nov 18 2010</div> <div class=sectline>Using (n, n+1) to generate a primitive Pythagorean triangle, the sides will be 2*n+1, 2*(n^2+n), and 2*n^2+2*n+1. Inscribing the largest rectangle with integral sides will have sides of length n and n^2+n. Side n is collinear to side 2*n+1 of the triangle and side n^2+n is collinear to side 2*(n^2+n) of the triangle. The areas of theses rectangles are a(n). - <a href="/wiki/User:J._M._Bergot">J. M. Bergot</a>, Sep 22 2011</div> <div class=sectline>a(n+1) is the sum of n-th row of the triangle in <a href="/A195437" title="Triangle formed by: 1 even, 2 odd, 3 even, 4 odd... starting with 2.">A195437</a>. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Nov 23 2011</div> <div class=sectline>Partial sums of <a href="/A049450" title="Pentagonal numbers multiplied by 2: a(n) = n*(3*n-1).">A049450</a>. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Jan 12 2013</div> <div class=sectline>From <a href="/wiki/User:Jon_Perry">Jon Perry</a>, May 11 2013: (Start)</div> <div class=sectline>Define a 'stable brick triangle' as:</div> <div class=sectline> -----</div> <div class=sectline> | c |</div> <div class=sectline> ---------</div> <div class=sectline> | a | | b |</div> <div class=sectline> ----------</div> <div class=sectline>with a, b, c > 0 and c <= a + b. This can be visualized as two bricks with a third brick on top. The third brick can only be as strong as a+b, otherwise the wall collapses - for example, (1,2,4) is unstable.</div> <div class=sectline>a(n) gives the number of stable brick triangles that can be formed if the two supporting bricks are 1 <= a <= n and 1 <= b <= n: a(n) = Sum_{a=1..n} Sum_{b=1..n} Sum_c 1 = n^3 + n^2 as given in the Adamchuk formula.</div> <div class=sectline>So for i=j=n=2 we have 4:</div> <div class=sectline> 1 2 3 4</div> <div class=sectline> 2 2 2 2 2 2 2 2</div> <div class=sectline>For example, n=2 gives 2 from [a=1,b=1], 3 from both [a=1,b=2] and [a=2,b=1] and 4 from [a=2,b=2] so a(2) = 2 + 3 + 3 + 4 = 12. (End)</div> <div class=sectline>Define the infinite square array m(n,k) by m(n,k) = (n-k)^2 if n >= k >= 0 and by m(n,k) = (k+n)*(k-n) if 0 <= n <= k. This contains <a href="/A120070" title="Triangle of numbers used to compute the frequencies of the spectral lines of the hydrogen atom.">A120070</a> below the diagonal. Then a(n) = Sum_{k=0..n} m(n,k) + Sum_{r=0..n} m(r,n), the "hook sum" of the terms to the left of m(n,n) and above m(n,n) with irrelevant (vanishing) terms on the diagonal. - <a href="/wiki/User:J._M._Bergot">J. M. Bergot</a>, Aug 16 2013</div> <div class=sectline>a(n) is the sum of all pairs with repetition drawn from the set of odd numbers 2*n-3. This is similar to <a href="/A027480" title="a(n) = n*(n+1)*(n+2)/2.">A027480</a> but using the odd integers instead. Example using n=3 gives the odd numbers 1,3,5: 1+1, 1+3, 1+5, 3+3, 3+5,5+5 having a total of 36=a(3). - <a href="/wiki/User:J._M._Bergot">J. M. Bergot</a>, Apr 05 2016</div> <div class=sectline>a(n) is the first Zagreb index of the complete graph K[n+1]. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph. - <a href="/wiki/User:Emeric_Deutsch">Emeric Deutsch</a>, Nov 07 2016</div> <div class=sectline>a(n-2) is the maximum sigma irregularity over all trees with n vertices. The extremal graphs are stars. (The sigma irregularity of a graph is the sum of squares of the differences between the degrees over all edges of the graph.) - <a href="/wiki/User:Allan_Bickle">Allan Bickle</a>, Jun 14 2023</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, pp. 50, 64.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Vincenzo Librandi, <a href="/A011379/b011379.txt">Table of n, a(n) for n = 0..1000</a></div> <div class=sectline>Bruno Berselli, A description of the recursive method in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).</div> <div class=sectline>Allan Bickle and Zhongyuan Che, <a href="https://doi.org/10.1016/j.dam.2023.01.020">Irregularities of Maximal k-degenerate Graphs</a>, Discrete Applied Math. 331 (2023) 70-87.</div> <div class=sectline>Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.</div> <div class=sectline>Ivan Gutman and Kinkar Ch. Das, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match50/match50_83-92.pdf">The first Zagreb index 30 years after</a>, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.</div> <div class=sectline>Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a></div> <div class=sectline><a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = 2*<a href="/A002411" title="Pentagonal pyramidal numbers: a(n) = n^2*(n+1)/2.">A002411</a>(n).</div> <div class=sectline>a(n) = Sum_{j=1..n} (Sum_{i=1..n} (i+j)), row sums of <a href="/A126890" title="Triangle read by rows: T(n,k) = n*(n+2*k+1)/2, 0 <= k <= n.">A126890</a> skipping numbers in the first column. - <a href="/wiki/User:Alexander_Adamchuk">Alexander Adamchuk</a>, Oct 12 2004</div> <div class=sectline>Sum_{n>0} 1/a(n) = (Pi^2 - 6)/6 = 0.6449340... [Jolley eq 272] - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Dec 22 2006</div> <div class=sectline>a(n) = 2*n*binomial(n+1,2) = 2*n*<a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(n). - <a href="/wiki/User:Arkadiusz_Wesolowski">Arkadiusz Wesolowski</a>, Feb 10 2012</div> <div class=sectline>G.f.: 2*x*(1 + 2*x)/(1 - x)^4. - <a href="/wiki/User:Arkadiusz_Wesolowski">Arkadiusz Wesolowski</a>, Feb 11 2012</div> <div class=sectline>a(n) = <a href="/A000330" title="Square pyramidal numbers: a(n) = 0^2 + 1^2 + 2^2 + ... + n^2 = n*(n+1)*(2*n+1)/6.">A000330</a>(n) + <a href="/A002412" title="Hexagonal pyramidal numbers, or greengrocer's numbers.">A002412</a>(n) = <a href="/A000292" title="Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.">A000292</a>(n) + <a href="/A002413" title="Heptagonal (or 7-gonal) pyramidal numbers: a(n) = n*(n+1)*(5*n-2)/6.">A002413</a>(n). - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Jan 11 2013</div> <div class=sectline>a(n) = <a href="/A245334" title="A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.">A245334</a>(n+1,2), n > 0. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Aug 31 2014</div> <div class=sectline>Sum_{n>=1} 1/a(n) = <a href="/A013661" title="Decimal expansion of Pi^2/6 = zeta(2) = Sum_{m>=1} 1/m^2.">A013661</a>-1. - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Oct 18 2019 [corrected by <a href="/wiki/User:Jason_Yuen">Jason Yuen</a>, Aug 04 2024]</div> <div class=sectline>Sum_{n>=1} (-1)^(n+1)/a(n) = 1 + Pi^2/12 - 2*log(2). - <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Jul 04 2020</div> <div class=sectline>E.g.f.: exp(x)*x*(2 + 4*x + x^2). - <a href="/wiki/User:Stefano_Spezia">Stefano Spezia</a>, May 20 2021</div> <div class=sectline>a(n) = n*<a href="/A002378" title="Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).">A002378</a>(n) = <a href="/A000578" title="The cubes: a(n) = n^3.">A000578</a>(n) + <a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>(n). - <a href="/wiki/User:J.S._Seneschal">J.S. Seneschal</a>, Jun 18 2024</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>a(3) = 3^2+3^3 = 36.</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline><a href="/A011379" title="a(n) = n^2*(n+1).">A011379</a>:=n->n^2*(n+1); seq(<a href="/A011379" title="a(n) = n^2*(n+1).">A011379</a>(n), n=0..40); # <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Feb 25 2014</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[n^3+n^2, {n, 0, 40}] (* <a href="/wiki/User:Vladimir_Joseph_Stephan_Orlovsky">Vladimir Joseph Stephan Orlovsky</a>, Jan 03 2009, modified by <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Aug 10 2019 *)</div> <div class=sectline>LinearRecurrence[{4, -6, 4, -1}, {0, 2, 12, 36}, 40] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Sep 13 2018 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Magma) [n^2+n^3: n in [0..40]]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, May 02 2011</div> <div class=sectline>(Haskell)</div> <div class=sectline>a011379 n = a000290 n + a000578 n -- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Apr 28 2013</div> <div class=sectline>(PARI) a(n)=n^3+n^2 \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Apr 06 2016</div> <div class=sectline>(Sage) [n^2*(n+1) for n in (0..40)] # <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Aug 10 2019</div> <div class=sectline>(GAP) List([0..40], n-> n^2*(n+1) ); # <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Aug 10 2019</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>, <a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>, <a href="/A000292" title="Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.">A000292</a>, <a href="/A000330" title="Square pyramidal numbers: a(n) = 0^2 + 1^2 + 2^2 + ... + n^2 = n*(n+1)*(2*n+1)/6.">A000330</a>, <a href="/A000578" title="The cubes: a(n) = n^3.">A000578</a>, <a href="/A002378" title="Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).">A002378</a>, <a href="/A002411" title="Pentagonal pyramidal numbers: a(n) = n^2*(n+1)/2.">A002411</a>, <a href="/A002412" title="Hexagonal pyramidal numbers, or greengrocer's numbers.">A002412</a>, <a href="/A002413" title="Heptagonal (or 7-gonal) pyramidal numbers: a(n) = n*(n+1)*(5*n-2)/6.">A002413</a>, <a href="/A005449" title="Second pentagonal numbers: a(n) = n*(3*n + 1)/2.">A005449</a>, <a href="/A013661" title="Decimal expansion of Pi^2/6 = zeta(2) = Sum_{m>=1} 1/m^2.">A013661</a>, <a href="/A022549" title="Sum of a square and a nonnegative cube.">A022549</a>, <a href="/A027480" title="a(n) = n*(n+1)*(n+2)/2.">A027480</a>, <a href="/A045991" title="a(n) = n^3 - n^2.">A045991</a>, <a href="/A049450" title="Pentagonal numbers multiplied by 2: a(n) = n*(3*n-1).">A049450</a>, <a href="/A120070" title="Triangle of numbers used to compute the frequencies of the spectral lines of the hydrogen atom.">A120070</a>, <a href="/A126890" title="Triangle read by rows: T(n,k) = n*(n+2*k+1)/2, 0 <= k <= n.">A126890</a>, <a href="/A195437" title="Triangle formed by: 1 even, 2 odd, 3 even, 4 odd... starting with 2.">A195437</a>, <a href="/A245334" title="A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.">A245334</a>.</div> <div class=sectline>Cf. <a href="/A011379" title="a(n) = n^2*(n+1).">A011379</a>, <a href="/A181617" title="Molecular topological indices of the complete graph K_n.">A181617</a>, <a href="/A270205" title="Number of 2 X 2 planar subsets in an n X n X n cube.">A270205</a> (sigma irregularities of maximal k-degenerate graphs).</div> <div class=sectline>Sequence in context: <a href="/A246426" title="Number of steps in base n after which all digits > 0 are present in x(i) where x(i) = x(i-1) + sum of missing digits of x(i-...">A246426</a> <a href="/A353503" title="Numbers whose product of prime indices equals their product of prime exponents (prime signature).">A353503</a> <a href="/A176583" title="n^2+Smallest cube, (Smallest cube >= n^2).">A176583</a> * <a href="/A338610" title="Integers m such that there exist one prime p and one positive integer k, for which the expression k^3 + k^2*p is a perfect c...">A338610</a> <a href="/A369175" title="Number of regions in a graph of n adjacent rectangles in a row with all possible diagonals drawn, as in A306302, but without...">A369175</a> <a href="/A073404" title="Coefficient triangle of polynomials (falling powers) related to convolutions of A002605(n), n>=0, (generalized (2,2)-Fibonac...">A073404</a></div> <div class=sectline>Adjacent sequences: <a href="/A011376" title="Decimal expansion of 17th root of 16.">A011376</a> <a href="/A011377" title="Expansion of 1/((1-x)*(1-2*x)*(1-x^2)).">A011377</a> <a href="/A011378" title="Decimal expansion of 19th root of 16.">A011378</a> * <a href="/A011380" title="Decimal expansion of 6th root of 17.">A011380</a> <a href="/A011381" title="Decimal expansion of 7th root of 17.">A011381</a> <a href="/A011382" title="Decimal expansion of 8th root of 17.">A011382</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>Glen Burch (gburch(AT)erols.com), <a href="/wiki/User:Felice_Russo">Felice Russo</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 April 10 12:28 EDT 2025. 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