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A006011
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n^2*(n^2-1)/4.
(Formerly M3044)
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16
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0, 0, 3, 18, 60, 150, 315, 588, 1008, 1620, 2475, 3630, 5148, 7098, 9555, 12600, 16320, 20808, 26163, 32490, 39900, 48510, 58443, 69828, 82800, 97500, 114075, 132678, 153468, 176610, 202275, 230640, 261888, 296208, 333795, 374850, 419580, 468198
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Products of two consecutive triangular numbers (A000217).
a(n) = number of Lyndon words of length 4 on an n-letter alphabet. A Lyndon word is a primitive word that is lexicographically smallest in its cyclic rotation class. For example, a(2)=3 counts 1112, 1122, 1222. - David Callan (callan(AT)stat.wisc.edu), Nov 29 2007
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REFERENCES
| S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
M. Azaola and F. Santos, The number of triangulations of the cyclic polytope C(n,n-4), Discrete Comput. Geom., 27 (2002), 29-48 (see Prop. 4.2(a)).
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FORMULA
| G.f.: 3*(1 + x ) / ( 1 - x )^5.
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 (alex(AT)kolmogorov.com), Apr 13 2006
a(n) = 3*A002415(n) = A047928(n-1)/4 = A083374(n-1)/2 = A008911(n)*3/2. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2007
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MAPLE
| [seq((binomial(3+n, 2)-binomial(2+n, 1))*(binomial(4+n, 3)-binomial(3+n, 3)), n=-2..39)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 29 2006
[seq (stirling2(n+1, n)*binomial(n, 2), n=0..37)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 05 2006
a:=n->sum(k^3+sum(k, k=0..n), k=0..n):seq(a(n), n=-1...36); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 01 2008
a:=n->sum(k^3+sum(k, k=0..n), k=0..n):seq(a(n), n=-1...36); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 09 2008]
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MATHEMATICA
| Table[ n^2*(n^2 - 1)/4, {n, 0, 38} ]
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PROG
| (MAGMA) [n^2*(n^2-1)/4: n in [0..40]]; // Vincenzo Librandi, Sep 14 2011
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CROSSREFS
| Thrice A002415. Row 4 of A074650.
Cf. A002415, A008911, A047928, A083374.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)
Equals for n=>2 second right hand column of A163932.
(End)
Sequence in context: A190313 A139362 A012763 * A012779 A074439 A000648
Adjacent sequences: A006008 A006009 A006010 * A006012 A006013 A006014
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
| More terms from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 29 2006
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