A176145 a(n) = n*(n-3)*(n^2-7*n+14)/8.

0, 1, 5, 18, 49, 110, 216, 385, 638, 999, 1495, 2156, 3015, 4108, 5474, 7155, 9196, 11645, 14553, 17974, 21965, 26586, 31900, 37973, 44874, 52675, 61451, 71280, 82243, 94424, 107910, 122791, 139160, 157113, 176749, 198170, 221481, 246790, 274208, 303849

Offset: 3

Author

Michel Lagneau, Apr 10 2010

Keywords

• nonn
• easy

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.

Examples

			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).
		

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).

Crossrefs

Cf. A080956, A055504.

Cf. A000217, A034856, A000124, A005581-A005584.

Programs

Magma

[n*(n-3)*(n^2 - 7*n + 14) / 8: n in [3..60]]; // Vincenzo Librandi, May 21 2011

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 *)

PARI

vector(100,n,(n+2)*(n-1)*(n^2-3*n+4)/8) \\ Derek Orr, Jan 21 2015

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

Extensions

Edited by N. J. A. Sloane, Apr 19 2010