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%I M3044 #111 Mar 12 2024 02:47:21
%S 0,0,3,18,60,150,315,588,1008,1620,2475,3630,5148,7098,9555,12600,
%T 16320,20808,26163,32490,39900,48510,58443,69828,82800,97500,114075,
%U 132678,153468,176610,202275,230640,261888,296208,333795,374850,419580,468198
%N a(n) = n^2*(n^2 - 1)/4.
%C Products of two consecutive triangular numbers (A000217).
%C 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 For n >= 2 this is the second rightmost column of A163932. - _Johannes W. Meijer_, Oct 16 2009
%C Partial sums of A059270. - _J. M. Bergot_, Jun 27 2013
%C 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 a(n) is the Wiener index of the triangular graph T(n+1). - _Emeric Deutsch_, Aug 26 2013
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A006011/b006011.txt">Table of n, a(n) for n = 0..10000</a>
%H 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 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: 3*(1 + x) / (1 - x)^5.
%F 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 a(n) = 3*A002415(n) = A047928(n-1)/4 = A083374(n-1)/2 = A008911(n)*3/2. - _Zerinvary Lajos_, May 09 2007
%F a(n) = (A126274(n) - A000537(n+1))/2. - _Enrique Pérez Herrero_, Mar 11 2013
%F Ceiling(sqrt(a(n)) + sqrt(a(n-1)))/2 = A000217(n). - _Richard R. Forberg_, Aug 14 2013
%F a(n) = Sum_{i=1..n-1} i*(i^2+n) for n > 1 (see Example section). - _Bruno Berselli_, Aug 29 2014
%F Sum_{n>=2} 1/a(n) = 7 - 2*Pi^2/3 = 0.42026373260709425411... . - _Vaclav Kotesovec_, Apr 27 2016
%F a(n) = A000217(n^2+n) - A000217(n)*A000217(n+1). - _Charlie Marion_, Feb 15 2020
%F Sum_{n>=2} (-1)^n/a(n) = Pi^2/3 - 3. - _Amiram Eldar_, Nov 02 2021
%F E.g.f.: exp(x)*x^2*(6 + 6*x + x^2)/4. - _Stefano Spezia_, Mar 12 2024
%e From _Bruno Berselli_, Aug 29 2014: (Start)
%e After the zeros, the sequence is provided by the row sums of the triangle:
%e 3;
%e 4, 14;
%e 5, 16, 39;
%e 6, 18, 42, 84;
%e 7, 20, 45, 88, 155;
%e 8, 22, 48, 92, 160, 258;
%e 9, 24, 51, 96, 165, 264, 399;
%e 10, 26, 54, 100, 170, 270, 406, 584;
%e 11, 28, 57, 104, 175, 276, 413, 592, 819;
%e 12, 30, 60, 108, 180, 282, 420, 600, 828, 1110; etc.,
%e where T(r,c) = c*(c^2+r+1), with r = row index, c = column index, r >= c > 0. (End)
%p A006011 := proc(n)
%p n^2*(n^2-1)/4 ;
%p end proc: # _R. J. Mathar_, Nov 29 2015
%t Table[n^2 (n^2 - 1)/4, {n, 0, 38}]
%t Binomial[Range[20]^2, 2]/2 (* _Eric W. Weisstein_, Sep 08 2017 *)
%t LinearRecurrence[{5, -10, 10, -5, 1}, {0, 3, 18, 60, 150}, 20] (* _Eric W. Weisstein_, Sep 08 2017 *)
%t CoefficientList[Series[-3 x (1 + x)/(-1 + x)^5, {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 08 2017 *)
%o (Magma) [n^2*(n^2-1)/4: n in [0..40]]; // _Vincenzo Librandi_, Sep 14 2011
%o (PARI) a(n)=binomial(n^2,2)/2 \\ _Charles R Greathouse IV_, Jun 27 2013
%Y Thrice A002415. Row 4 of A074650.
%Y Cf. A000217, A000290, A000537, A001477, A002415, A008911, A047928, A059270, A083374, A126274, A163932, A228317
%Y A column of A124428.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, _Simon Plouffe_
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