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A007290 a(n) = 2*binomial(n,3).
(Formerly M1831)
52

%I M1831

%S 0,0,0,2,8,20,40,70,112,168,240,330,440,572,728,910,1120,1360,1632,

%T 1938,2280,2660,3080,3542,4048,4600,5200,5850,6552,7308,8120,8990,

%U 9920,10912,11968,13090,14280,15540,16872,18278,19760,21320,22960

%N a(n) = 2*binomial(n,3).

%C Number of acute triangles made from the vertices of a regular n-polygon when n is even (cf. A000330). - _Sen-Peng Eu_, Apr 05 2001

%C a(n+2)=(-1)*coefficient of X in Zagier's polynomial (n,n-1). - _Benoit Cloitre_, Oct 12 2002

%C Definite integrals of certain products of 2 derivatives of (orthogonal) Chebyshev polynomials of the 2nd kind are pi-multiple of this sequence. For even (p+q): Integrate[ D[ChebyshevU[p, x], x] D[ChebyshevU[q, x], x] (1 - x^2)^(1/2), {x,-1,1}] / Pi = a(n), where n=Min[p,q]. Example: a(3)=20 because Integrate[ D[ChebyshevU[3, x], x] D[ChebyshevU[5, x], x] (1 - x^2)^(1/2), {x,-1,1}]/Pi = 20 since 3=Min[3,5] and 3+5 is even. - Christoph Pacher (Christoph.Pacher(AT)arcs.ac.at), Dec 16 2004

%C If Y is a 2-subset of an n-set X then, for n>=3, a(n-1) is the number of 3-subsets and 4-subsets of X having exactly one element in common with Y. - _Milan Janjic_, Dec 28 2007

%C a(n) is also the number of proper colorings of the cycle graph Csub3 (also the complete graph Ksub3) when n colors are available. - _Gary E. Stevens_, Dec 28 2008

%C a(n) is the reverse Wiener index of the path graph with n vertices. See the Balaban et al. reference, p. 927.

%C For n > 1: a(n) = sum of (n-1)-th row of A141418. - _Reinhard Zumkeller_, Nov 18 2012

%C This is the sequence for nuclear magic numbers in an idealized spherical nucleus under the harmonic oscillator model. - _Jess Tauber_, May 20 2013

%C Sum(n>2, 1/a(n)) = 3/4. - _Enrique Pérez Herrero_, Nov 10 2013

%C Shifted non-vanishing diagonal of A132440^3/3. Second subdiagonal of A238363 (without zeros). For n>0, a(n+2)=n*(n+1)*(n+2)/3. Cf. A130534 for relations to colored forests and disposition of flags on flagpoles. - _Tom Copeland_, Apr 05 2014

%C a(n) is the number of ordered rooted trees with n non-root nodes that have 2 leaves; see A108838. - _Joerg Arndt_, Aug 18 2014

%D L. Berzolari, Allgemeine Theorie der Höheren Ebenen Algebraischen Kurven, Encyclopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. Band III_2. Heft 3, Leipzig: B. G. Teubner, 1906. p. 352.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A007290/b007290.txt">Table of n, a(n) for n = 0..1000</a>

%H A. T. Balaban, D. Mills, O. Ivanciuc, and S. C. Basak, <a href="http://public.carnet.hr/ccacaa/CCA-PDF/cca2000/v73-n4/cca_73_2000_923-941_Balaban.pdf">Reverse Wiener indices</a>, Croatica Chemica Acta, 73 (4), 2000, 923-941.

%H A. Burstein, S. Kitaev, T. Mansour, <a href="http://puma.dimai.unifi.it/19_2_3/3.pdf">Partially ordered patterns and their combinatorial interpretations</a>, PU. M. A. Vol. 19 (2008), No. 2-3, pp. 27-38.

%H O. Haxel et al., <a href="http://dx.doi.org/10.1103/PhysRev.75.1766.2">On the "Magic Numbers" in Nuclear Structure</a>, Phys. Rev., 75 (1949), 1766.

%H Xiangdong Ji, <a href="http://www.physics.umd.edu/courses/Phys741/xji/chap8_12.pdf">Chapter 8: Structure of Finite Nuclei</a>, Lecture notes for Phys 741 at Univ. of Maryland, p. 140 [From _Tom Copeland_, Apr 07 2014].

%H V. Ladma, <a href="http://www.sweb.cz/vladimir_ladma/english/notes/texts/magicn.htm">Magic Numbers</a>

%H Hamzeh Mujahed and Benedek Nagy, <a href="http://link.springer.com/chapter/10.1007/978-3-319-18720-4_50#page-1">Wiener Index on Lines of Unit Cells of the Body-Centered Cubic Grid</a>, Mathematical Morphology and Its Applications to Signal and Image Processing, 12th International Symposium, ISMM 2015.

%H V. B. Priezzhev, <a href="http://dx.doi.org/10.1088/0305-4470/12/11/022">Series expansion for rectilinear polymers on the square lattice</a>, J. Phys. A 12 (1979), 2131-2139.

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F G.f.: 2*x^3/(1-x)^4.

%F a(n) = a(n-1)*n/(n-3) = a(n-1)+A002378(n-2) = 2*A000292(n-2) = Sum_{i = 0 to n-2} (i*(i+1)) = n(n-1)(n-2)/3. - _Henry Bottomley_, Jun 02 2000 [Formula corrected by _R. J. Mathar_, Dec 13 2010]

%F a(n) = A000217(n-2) + A000330(n-2), n>1. - _Reinhard Zumkeller_, Mar 20 2008

%F a(n+1) = A000330(n) - A000217(n), n>=0. - _Zak Seidov_, Aug 07 2010

%F a(n) = A033487(n-2) - A052149(n-1) for n>1. - _Bruno Berselli_, Dec 10 2010

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 19 2012

%F a(n) = (2*n - 3*n^2 + n^3)/3. - _T. D. Noe_, May 20 2013

%F a(n+1) = A002412(n) - A000330(n) or "Hex Pyramidal" - "Square Pyramidal" (as can also be seen via above formula). - _Richard R. Forberg_, Aug 07 2013

%F E.g.f.: exp(x)*x^3/3. - _Geoffrey Critzer_, Nov 22 2015

%p A007290 := proc(n) 2*binomial(n,3) ; end proc:

%p a:=n-> sum(i^2-i, i=0..n-1): seq(a(n), n=0..41); # _Zerinvary Lajos_, Apr 25 2008

%t Table[Integrate[ D[ChebyshevU[n, x], x] D[ChebyshevU[n, x], x] (1 - x^2)^(1/2), {x, -1, 1}]/Pi, {n, 1, 20}] (* Pacher *)

%t lst={0};s=0;Do[s+=n^2-n;AppendTo[lst, s], {n, 0, 5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Sep 30 2008 *)

%t Table[Sum[i^2 + i, {i, 0, n - 1}], {n, -1, 40}] (* _Zerinvary Lajos_, Jul 10 2009 *)

%t LinearRecurrence[{4,-6,4,-1},{0,0,0,2},50] (* _Vincenzo Librandi_, Jun 19 2012 *)

%o (MAGMA) I:=[0, 0, 0, 2]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // _Vincenzo Librandi_, Jun 19 2012

%o (Haskell)

%o a007290 n = if n < 3 then 0 else 2 * a007318 n 3

%o -- _Reinhard Zumkeller_, Nov 18 2012

%o (PARI) x='x+O('x^100); concat([0, 0, 0], Vec(2*x^3/(1-x)^4)) \\ _Altug Alkan_, Nov 01 2015

%Y A diagonal of A059419. Partial sums of A002378.

%Y A diagonal of A008291. Row 3 of A074650.

%Y Cf. A145066, A051925, A145067, A145068.

%Y Cf. A210569.

%K nonn,easy

%O 0,4

%A _N. J. A. Sloane_, _Simon Plouffe_

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Last modified December 4 06:30 EST 2016. Contains 278749 sequences.