The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A152618 a(n) = (n-1)^2*(n+1). 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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 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 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 *) PROG (Magma) [(n-1)^2*(n+1): n in [0..50]]; // Vincenzo Librandi, Jun 25 2013 (PARI) a(n)=(n+1)*(n-1)^2 \\ Charles R Greathouse IV, Mar 21 2014 CROSSREFS Sequence in context: A271374 A147874 A092466 * A296947 A255211 A172482 Adjacent sequences: A152615 A152616 A152617 * A152619 A152620 A152621 KEYWORD nonn,easy AUTHOR Philippe Deléham, Dec 09 2008 STATUS approved

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

Last modified November 30 11:38 EST 2022. Contains 358441 sequences. (Running on oeis4.)