

A176145


a(n) = n*(n3)*(n^27*n+14)/8.


6



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
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OFFSET

3,3


COMMENTS

Number of points of intersection of diagonals of a general convex npolygon. (both inside and outside the polygon).
n(n3)/2 (n >= 3) is the number of diagonals of an ngon (A080956). The number of points (inside or outside), distinct of tops, where these diagonals intersect is : (1/2)( n(n3)/2)(n(n3)/2  2(n4) 1) = n(n3)(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


FORMULA

G.f.: x^4*(1+3*x^2x^3)/(1x)^5. [Colin Barker, Jan 31 2012]
a(n) = 5*a(n1) 10*a(n2) +10*a(n3) 5*a(n4) + a(n5), 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=(n3)..(n2)*(n3)/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*(n3)*(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 *)


PROG

(MAGMA) [n*(n3)*(n^2  7*n + 14) / 8: n in [3..60]]; // Vincenzo Librandi, May 21 2011
(PARI) vector(100, n, (n+2)*(n1)*(n^23*n+4)/8) \\ Derek Orr, Jan 21 2015


CROSSREFS

Cf. A080956, A055504.
Cf. A000217, A034856, A000124, A005581A005584.
Sequence in context: A109363 A218214 A146213 * A270978 A272512 A257055
Adjacent sequences: A176142 A176143 A176144 * A176146 A176147 A176148


KEYWORD

nonn,easy


AUTHOR

Michel Lagneau, Apr 10 2010


EXTENSIONS

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


STATUS

approved



