login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A007290 a(n) = 2*binomial(n,3).
(Formerly M1831)
52
0, 0, 0, 2, 8, 20, 40, 70, 112, 168, 240, 330, 440, 572, 728, 910, 1120, 1360, 1632, 1938, 2280, 2660, 3080, 3542, 4048, 4600, 5200, 5850, 6552, 7308, 8120, 8990, 9920, 10912, 11968, 13090, 14280, 15540, 16872, 18278, 19760, 21320, 22960 (list; graph; refs; listen; history; text; internal format)
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)=(-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

Sum(n>2, 1/a(n)) = 3/4. - Enrique Pérez Herrero, Nov 10 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

REFERENCES

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.

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

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

A. T. Balaban, D. Mills, O. Ivanciuc, and S. C. Basak, Reverse Wiener indices, Croatica Chemica Acta, 73 (4), 2000, 923-941.

A. Burstein, S. Kitaev, T. Mansour, Partially ordered patterns and their combinatorial interpretations, PU. M. A. Vol. 19 (2008), No. 2-3, pp. 27-38.

O. Haxel et al., On the "Magic Numbers" in Nuclear Structure, Phys. Rev., 75 (1949), 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].

V. Ladma, Magic Numbers

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 12 (1979), 2131-2139.

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.

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 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]

a(n) = A000217(n-2) + A000330(n-2), n>1. - Reinhard Zumkeller, Mar 20 2008

a(n+1) = A000330(n) - A000217(n), n>=0. - Zak Seidov, Aug 07 2010

a(n) = A033487(n-2) - A052149(n-1) for n>1. - Bruno Berselli, Dec 10 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

E.g.f.: exp(x)*x^3/3. - Geoffrey Critzer, Nov 22 2015

MAPLE

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

a:=n-> sum(i^2-i, i=0..n-1): seq(a(n), n=0..41); # Zerinvary Lajos, Apr 25 2008

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 *)

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

Table[Sum[i^2 + i, {i, 0, n - 1}], {n, -1, 40}] (* Zerinvary Lajos, Jul 10 2009 *)

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) x='x+O('x^100); concat([0, 0, 0], Vec(2*x^3/(1-x)^4)) \\ Altug Alkan, Nov 01 2015

CROSSREFS

A diagonal of A059419. Partial sums of A002378.

A diagonal of A008291. Row 3 of A074650.

Cf. A145066, A051925, A145067, A145068.

Cf. A210569.

Sequence in context: A025219 A032767 A032633 * A049031 A058037 A203420

Adjacent sequences:  A007287 A007288 A007289 * A007291 A007292 A007293

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Simon Plouffe

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 22 15:30 EDT 2017. Contains 286876 sequences.