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%I M1831 #226 Aug 03 2024 01:49:12
%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 9920,10912,11968,13090,14280,15540,16872,18278,19760,21320,22960,24682,26488,28380,30360,32430,34592,36848,39200
%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) is (-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 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
%C Number of floating point multiplications in the factorization of an (n-1)X(n-1) real matrix by Gaussian elimination as e.g. implemented in LINPACK subroutines sgefa.f or dgefa.f. The number of additions is given by A000330. - _Hugo Pfoertner_, Mar 28 2018
%C a(n+1) = Max_{s in S_n} Sum_{k=1..n} (k - s(k))^2 where S_n is the symmetric group of permutations of [1..n]; this maximum is obtained with the permutation s = (1, n) (2, n-1) (3, n-2) ... (k, n-k+1). (see Protat reference). - _Bernard Schott_, Dec 26 2022
%D Luigi 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 Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 259.
%D Maurice Protat, Des Olympiades à l'Agrégation, un problème de maximum, Problème 36, p. 83, Ellipses, Paris 1997.
%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 Alexandru T. Balaban, Denise Mills, Ovidiu Ivanciuc and Subhash C. Basak,, <a href="https://hrcak.srce.hr/file/194901">Reverse Wiener indices</a>, Croatica Chemica Acta, Vol. 73, No. 4 (2000), pp. 923-941.
%H A. Burstein, S. Kitaev and 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, No. 2-3 (2008), pp. 27-38.
%H Otto Haxel, J. Hans D. Jensen and Hans E. Suess, <a href="http://dx.doi.org/10.1103/PhysRev.75.1766.2">On the "Magic Numbers" in Nuclear Structure</a>, Phys. Rev., Vol. 75 (1949), p. 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 Sandi Klavžar, Balázs Patkós, Gregor Rus and Ismael G. Yero, <a href="https://arxiv.org/abs/1907.04535">On general position sets in Cartesian grids</a>, arXiv:1907.04535 [math.CO], 2019.
%H Vladimir Ladma, <a href="http://www.traced-ideas.eu/atom/atomcore.html">Magic Numbers</a>.
%H Cleve Moler, <a href="http://www.netlib.org/linpack/sgefa.f">LINPACK subroutine sgefa.f</a>, University of New Mexico, Argonne National Lab, 1978.
%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, Vol. 12, No. 11 (1979), pp. 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 Wikipedia, <a href="http://en.wikipedia.org/wiki/P-derivation">p-derivation</a>.
%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..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 Sum_{n>=3} 1/a(n) = 3/4. - _Enrique Pérez Herrero_, Nov 10 2013
%F E.g.f.: exp(x)*x^3/3. - _Geoffrey Critzer_, Nov 22 2015
%F a(n+2) = delta(-n) = -delta(n) for n >= 0, where delta is the p-derivation over the integers with respect to prime p = 3. - _Danny Rorabaugh_, Nov 10 2017
%F (a(n) + a(n+1))/2 = A000330(n-1). - _Ezhilarasu Velayutham_, Apr 05 2019
%F Sum_{n>=3} (-1)^(n+1)/a(n) = 6*log(2) - 15/4. - _Amiram Eldar_, Jan 09 2022
%F a(n) = Sum_{m=0..n-2} Sum_{k=0..n-2} abs(m-k). - _Nicolas Bělohoubek_, Nov 06 2022
%F From _Bernard Schott_, Jan 04 2023: (Start)
%F a(n) = 2 * A000292(n-2), for n >= 2.
%F a(n+1) = 2 *Sum_{k=1..floor(n/2)} (n-(2k-1))^2, for n >= 2. (End)
%p A007290 := proc(n) 2*binomial(n,3) end proc:
%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 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) a007290 n = if n < 3 then 0 else 2 * a007318 n 3 -- _Reinhard Zumkeller_, Nov 18 2012
%o (PARI) my(x='x+O('x^100)); concat([0, 0, 0], Vec(2*x^3/(1-x)^4)) \\ _Altug Alkan_, Nov 01 2015
%o (PARI) apply( {A007290(n)=binomial(n,3)*2}, [0..55]) \\ _M. F. Hasler_, Jul 02 2021
%Y A diagonal of A059419. Partial sums of A002378.
%Y A diagonal of A008291. Row 3 of A074650.
%Y Cf. A000292, A051925, A145066, A145067, A145068, A210569.
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_, _Simon Plouffe_
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