OFFSET
0,4
COMMENTS
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
a(n+2) is (-1)*coefficient of X in Zagier's polynomial (n,n-1). - Benoit Cloitre, Oct 12 2002
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
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
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
a(n) is the reverse Wiener index of the path graph with n vertices. See the Balaban et al. reference, p. 927.
For n > 1: a(n) = sum of (n-1)-th row of A141418. - Reinhard Zumkeller, Nov 18 2012
This is the sequence for nuclear magic numbers in an idealized spherical nucleus under the harmonic oscillator model. - Jess Tauber, May 20 2013
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
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
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
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
REFERENCES
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.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 259.
Maurice Protat, Des Olympiades à l'Agrégation, un problème de maximum, Problème 36, p. 83, Ellipses, Paris 1997.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alexandru T. Balaban, Denise Mills, Ovidiu Ivanciuc and Subhash C. Basak,, Reverse Wiener indices, Croatica Chemica Acta, Vol. 73, No. 4 (2000), pp. 923-941.
A. Burstein, S. Kitaev and T. Mansour, Partially ordered patterns and their combinatorial interpretations, PU. M. A. Vol. 19, No. 2-3 (2008), pp. 27-38.
Otto Haxel, J. Hans D. Jensen and Hans E. Suess, On the "Magic Numbers" in Nuclear Structure, Phys. Rev., Vol. 75 (1949), p. 1766.
Xiangdong Ji, Chapter 8: Structure of Finite Nuclei, Lecture notes for Phys 741 at Univ. of Maryland, p. 140 [From Tom Copeland, Apr 07 2014].
Sandi Klavžar, Balázs Patkós, Gregor Rus and Ismael G. Yero, On general position sets in Cartesian grids, arXiv:1907.04535 [math.CO], 2019.
Vladimir Ladma, Magic Numbers.
Cleve Moler, LINPACK subroutine sgefa.f, University of New Mexico, Argonne National Lab, 1978.
Hamzeh Mujahed and Benedek Nagy, Wiener Index on Lines of Unit Cells of the Body-Centered Cubic Grid, Mathematical Morphology and Its Applications to Signal and Image Processing, 12th International Symposium, ISMM 2015.
V. B. Priezzhev, Series expansion for rectilinear polymers on the square lattice, J. Phys. A, Vol. 12, No. 11 (1979), pp. 2131-2139.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Wikipedia, p-derivation.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
G.f.: 2*x^3/(1-x)^4.
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]
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 19 2012
a(n) = (2*n - 3*n^2 + n^3)/3. - T. D. Noe, May 20 2013
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
Sum_{n>=3} 1/a(n) = 3/4. - Enrique Pérez Herrero, Nov 10 2013
E.g.f.: exp(x)*x^3/3. - Geoffrey Critzer, Nov 22 2015
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
(a(n) + a(n+1))/2 = A000330(n-1). - Ezhilarasu Velayutham, Apr 05 2019
Sum_{n>=3} (-1)^(n+1)/a(n) = 6*log(2) - 15/4. - Amiram Eldar, Jan 09 2022
a(n) = Sum_{m=0..n-2} Sum_{k=0..n-2} abs(m-k). - Nicolas Bělohoubek, Nov 06 2022
From Bernard Schott, Jan 04 2023: (Start)
a(n) = 2 * A000292(n-2), for n >= 2.
a(n+1) = 2 *Sum_{k=1..floor(n/2)} (n-(2k-1))^2, for n >= 2. (End)
MAPLE
A007290 := proc(n) 2*binomial(n, 3) end proc:
MATHEMATICA
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 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 0, 0, 2}, 50] (* Vincenzo Librandi, Jun 19 2012 *)
PROG
(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
(Haskell) a007290 n = if n < 3 then 0 else 2 * a007318 n 3 -- Reinhard Zumkeller, Nov 18 2012
(PARI) my(x='x+O('x^100)); concat([0, 0, 0], Vec(2*x^3/(1-x)^4)) \\ Altug Alkan, Nov 01 2015
(PARI) apply( {A007290(n)=binomial(n, 3)*2}, [0..55]) \\ M. F. Hasler, Jul 02 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved