OFFSET
3,3
COMMENTS
Number of points of intersection of diagonals of a general convex n-polygon. (both inside and outside the polygon).
n(n-3)/2 (n >= 3) is the number of diagonals of an n-gon (A080956). The number of points (inside or outside), distinct of tops, where these diagonals intersect is : (1/2)( n(n-3)/2)(n(n-3)/2 - 2(n-4) -1) = n(n-3)(n^2 - 7n + 14) / 8.
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 797.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 3..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
G.f.: x^4*(1+3*x^2-x^3)/(1-x)^5. [Colin Barker, Jan 31 2012]
a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) + a(n-5), with a(3)= 0, a(4)= 1, a(5)=5, a(6)= 18, a(7) = 49. [Bobby Milazzo, Jun 24 2013]
a(n) = Sum_{k=(n-3)..(n-2)*(n-3)/2} k. - J. M. Bergot, Jan 21 2015
EXAMPLE
For n=3, a(3) = 0 (no diagonals in triangle),
For n=4, a(4) = 1 (2 diagonals => 1 point of intersection),
For n=5, a(5) = 5 (5 diagonals => 5 points of intersection),
For n=6, a(6) = 18 (9 diagonals => 18 points of intersection).
MAPLE
for n from 3 to 50 do: x:=n*(n-3)*(n^2 - 7*n +14)/8 : print(x):od:
MATHEMATICA
Table[n*(n - 3)*(n^2 - 7*n + 14)/8, {n, 3, 42}] (* Bobby Milazzo, Jun 24 2013 *)
Drop[CoefficientList[Series[x^4(1+3x^2-x^3)/(1-x)^5, {x, 0, 50}], x], 3] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 5, 18, 49}, 50] (* Harvey P. Dale, Mar 14 2022 *)
PROG
(Magma) [n*(n-3)*(n^2 - 7*n + 14) / 8: n in [3..60]]; // Vincenzo Librandi, May 21 2011
(PARI) vector(100, n, (n+2)*(n-1)*(n^2-3*n+4)/8) \\ Derek Orr, Jan 21 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michel Lagneau, Apr 10 2010
EXTENSIONS
Edited by N. J. A. Sloane, Apr 19 2010
STATUS
approved