A152618 a(n) = (n-1)^2*(n+1).

1, 0, 3, 16, 45, 96, 175, 288, 441, 640, 891, 1200, 1573, 2016, 2535, 3136, 3825, 4608, 5491, 6480, 7581, 8800, 10143, 11616, 13225, 14976, 16875, 18928, 21141, 23520, 26071, 28800, 31713, 34816, 38115, 41616, 45325, 49248, 53391, 57760, 62361

Offset: 0

Author

Philippe Deléham, Dec 09 2008

Keywords

• nonn
• easy

Comments

For n>0 this is the same under substitution of variables as d(d-2)^2, the number of connected components in Bertrand et al.: "We construct a polynomial of degree d in two variables whose Hessian curve has (d-2)^2 connected components using Viro patchworking. In particular, this implies the existence of a smooth real algebraic surface of degree d in RP^3 whose parabolic curve is smooth and has d(d-2)^2 connected components." - Jonathan Vos Post, Apr 30 2009

For n>0 a(n) is twice the area of the trapezoid created by plotting the four points (n-1,n), (n,n-1), (n*(n-1)/2,n*(n+1)/2), (n*(n+1)/2,(n-1)*n/2). - J. M. Bergot, Mar 22 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Benoît Bertand and Erwan Brugallé, On the number of connected components of the parabolic curve, Comptes Rendus Mathématique, Vol. 348, No. 5-6 (2010), pp. 287-289; arXiv preprint, arXiv:0904.4652 [math.AG], Apr 29 2009. - _Jonathan Vos Post_, Apr 30 2009
• Jim Propp and Adam Propp-Gubin, Counting Triangles in Triangles, arXiv:2409.17117 [math.CO], 2024. See p. 9.

Programs

Magma

[(n-1)^2*(n+1): n in [0..50]]; // Vincenzo Librandi, Jun 25 2013

Maple

A152618:=n->(n-1)^2*(n+1); seq(A152618(k), k=0..100); # Wesley Ivan Hurt, Oct 06 2013

Mathematica

f[n_]:=(n-1)^2*(n+1);f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011*)
CoefficientList[Series[(9 x^2 - 4 x + 1)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 25 2013 *)

PARI

a(n)=(n+1)*(n-1)^2 \\ Charles R Greathouse IV, Mar 21 2014

Formula

a(n) = n^3 - n^2 - n + 1 = A083074(n) + 2. - Jeremy Gardiner, Jun 23 2013

G.f.: (9*x^2 - 4*x + 1)/(1-x)^4. - Vincenzo Librandi, Jun 25 2013

a(n+1) = A005449(n) + A002414(n), n > 0. - Wesley Ivan Hurt, Oct 06 2013

Sum_{n>1} 1/a(n) = (1/24) * (2*Pi^2 - 9). - Enrique Pérez Herrero, May 31 2015

Sum_{n>=2} (-1)^n/a(n) = (Pi^2 - 3)/24. - Amiram Eldar, Jan 13 2021

E.g.f.: exp(x)*(x^3+2*x^2-x+1). - Nikolaos Pantelidis, Feb 06 2023