cp's OEIS Frontend

A006011 a(n) = n^2*(n^2 - 1)/4.

%I A006011 M3044 #118 Aug 08 2025 17:44:37
%S A006011 0,0,3,18,60,150,315,588,1008,1620,2475,3630,5148,7098,9555,12600,
%T A006011 16320,20808,26163,32490,39900,48510,58443,69828,82800,97500,114075,
%U A006011 132678,153468,176610,202275,230640,261888,296208,333795,374850,419580,468198
%N A006011 a(n) = n^2*(n^2 - 1)/4.
%C A006011 Products of two consecutive triangular numbers (A000217).
%C A006011 a(n) is the number of Lyndon words of length 4 on an n-letter alphabet. A Lyndon word is a primitive word that is lexicographically earliest in its cyclic rotation class. For example, a(2)=3 counts 1112, 1122, 1222. - _David Callan_, Nov 29 2007
%C A006011 For n >= 2 this is the second rightmost column of A163932. - _Johannes W. Meijer_, Oct 16 2009
%C A006011 Partial sums of A059270. - _J. M. Bergot_, Jun 27 2013
%C A006011 Using the integers, triangular numbers, and squares plot the points (A001477(n),A001477(n+1)), (A000217(n), A000217(n+1)), and (A000290(n),A000290(n+1)) to create the vertices of a triangle. One-half the area of this triangle = a(n). - _J. M. Bergot_, Aug 01 2013
%C A006011 a(n) is the Wiener index of the triangular graph T(n+1). - _Emeric Deutsch_, Aug 26 2013
%D A006011 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006011 Vincenzo Librandi, <a href="/A006011/b006011.txt">Table of n, a(n) for n = 0..10000</a>
%H A006011 Miguel Azaola and Francisco Santos, <a href="http://personales.unican.es/santosf/Articulos/numcyclic.pdf">The number of triangulations of the cyclic polytope C(n,n-4)</a>, Discrete Comput. Geom., Vol. 27 (2002), pp. 29-48 (see Prop. 4.2(a)).
%H A006011 S. M. Losanitsch, <a href="http://dx.doi.org/10.1002/cber.189703002144">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber., Vol. 30 (1897), pp. 1917-1926.
%H A006011 S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber., Vol. 30 (1897), pp. 1917-1926. (Annotated scanned copy)
%H A006011 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>.
%H A006011 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>.
%H A006011 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A006011 G.f.: 3*(1 + x) / (1 - x)^5.
%F A006011 a(n) = (n-1)*n/2 * n*(n+1)/2 = A000217(n-1)*A000217(n) = 1/2*(n^2-1)*n^2/2 = 1/2*A000217(n^2-1). - _Alexander Adamchuk_, Apr 13 2006
%F A006011 a(n) = 3*A002415(n) = A047928(n-1)/4 = A083374(n-1)/2 = A008911(n)*3/2. - _Zerinvary Lajos_, May 09 2007
%F A006011 a(n) = (A126274(n) - A000537(n+1))/2. - _Enrique Pérez Herrero_, Mar 11 2013
%F A006011 Ceiling(sqrt(a(n)) + sqrt(a(n-1)))/2 = A000217(n). - _Richard R. Forberg_, Aug 14 2013
%F A006011 a(n) = Sum_{i=1..n-1} i*(i^2+n) for n > 1 (see Example section). - _Bruno Berselli_, Aug 29 2014
%F A006011 Sum_{n>=2} 1/a(n) = 7 - 2*Pi^2/3 = 0.42026373260709425411... . - _Vaclav Kotesovec_, Apr 27 2016
%F A006011 a(n) = A000217(n^2+n) - A000217(n)*A000217(n+1). - _Charlie Marion_, Feb 15 2020
%F A006011 Sum_{n>=2} (-1)^n/a(n) = Pi^2/3 - 3. - _Amiram Eldar_, Nov 02 2021
%F A006011 E.g.f.: exp(x)*x^2*(6 + 6*x + x^2)/4. - _Stefano Spezia_, Mar 12 2024
%e A006011 From _Bruno Berselli_, Aug 29 2014: (Start)
%e A006011 After the zeros, the sequence is provided by the row sums of the triangle:
%e A006011    3;
%e A006011    4, 14;
%e A006011    5, 16, 39;
%e A006011    6, 18, 42,  84;
%e A006011    7, 20, 45,  88, 155;
%e A006011    8, 22, 48,  92, 160, 258;
%e A006011    9, 24, 51,  96, 165, 264, 399;
%e A006011   10, 26, 54, 100, 170, 270, 406, 584;
%e A006011   11, 28, 57, 104, 175, 276, 413, 592, 819;
%e A006011   12, 30, 60, 108, 180, 282, 420, 600, 828, 1110; etc.,
%e A006011 where T(r,c) = c*(c^2+r+1), with r = row index, c = column index, r >= c > 0. (End)
%p A006011 A006011 := proc(n)
%p A006011     n^2*(n^2-1)/4 ;
%p A006011 end proc: # _R. J. Mathar_, Nov 29 2015
%t A006011 Table[n^2 (n^2 - 1)/4, {n, 0, 38}]
%t A006011 Binomial[Range[20]^2, 2]/2 (* _Eric W. Weisstein_, Sep 08 2017 *)
%t A006011 LinearRecurrence[{5, -10, 10, -5, 1}, {0, 3, 18, 60, 150}, 20] (* _Eric W. Weisstein_, Sep 08 2017 *)
%t A006011 CoefficientList[Series[-3 x (1 + x)/(-1 + x)^5, {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 08 2017 *)
%t A006011 Join[{0},Times@@@Partition[Accumulate[Range[0,40]],2,1]] (* _Harvey P. Dale_, Aug 08 2025 *)
%o A006011 (Magma) [n^2*(n^2-1)/4: n in [0..40]]; // _Vincenzo Librandi_, Sep 14 2011
%o A006011 (PARI) a(n)=binomial(n^2,2)/2 \\ _Charles R Greathouse IV_, Jun 27 2013
%Y A006011 Thrice A002415. Row 4 of A074650.
%Y A006011 Cf. A000217, A000290, A000537, A001477, A002415, A008911, A047928, A059270, A083374, A126274, A163932, A228317
%Y A006011 A column of A124428.
%K A006011 nonn,easy
%O A006011 0,3
%A A006011 _N. J. A. Sloane_, _Simon Plouffe_