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A152618 a(n) = (n-1)^2*(n+1). 7

%I

%S 1,0,3,16,45,96,175,288,441,640,891,1200,1573,2016,2535,3136,3825,

%T 4608,5491,6480,7581,8800,10143,11616,13225,14976,16875,18928,21141,

%U 23520,26071,28800,31713,34816,38115,41616,45325,49248,53391,57760,62361

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

%C 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

%C For n>0 a(n)=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

%H Vincenzo Librandi, <a href="/A152618/b152618.txt">Table of n, a(n) for n = 0..1000</a>

%H Benoit Bertrand, Erwan Brugallé, <a href="http://arxiv.org/abs/0904.4652">On the number of connected components of the parabolic curve</a>, arXiv:0904.4652 [math.AG], Apr 29 2009. - _Jonathan Vos Post_, Apr 30 2009

%F a(n) = n^3 - n^2 - n + 1 = A083074(n) + 2. - _Jeremy Gardiner_, Jun 23 2013

%F G.f.: (9*x^2 - 4*x + 1)/(1-x)^4. - _Vincenzo Librandi_, Jun 25 2013

%F a(n+1) = A005449(n) + A002414(n), n > 0. - _Wesley Ivan Hurt_, Oct 06 2013

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

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

%t f[n_]:=(n-1)^2*(n+1);f[Range[0,60]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 05 2011*)

%t CoefficientList[Series[(9 x^2 - 4 x + 1)/(1 - x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 25 2013 *)

%o (MAGMA) [(n-1)^2*(n+1): n in [0..50]]; // _Vincenzo Librandi_, Jun 25 2013

%o (PARI) a(n)=(n+1)*(n-1)^2 \\ _Charles R Greathouse IV_, Mar 21 2014

%K nonn,easy

%O 0,3

%A _Philippe Deléham_, Dec 09 2008

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Last modified September 20 21:19 EDT 2019. Contains 327247 sequences. (Running on oeis4.)