|
|
A152618
|
|
a(n) = (n-1)^2*(n+1).
|
|
9
|
|
|
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
|
|
|
FORMULA
|
Sum_{n>=2} (-1)^n/a(n) = (Pi^2 - 3)/24. - Amiram Eldar, Jan 13 2021
|
|
MAPLE
|
|
|
MATHEMATICA
|
CoefficientList[Series[(9 x^2 - 4 x + 1)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 25 2013 *)
|
|
PROG
|
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|