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A176145 a(n) = n*(n-3)*(n^2-7*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 (list; graph; refs; listen; history; text; internal format)
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

Cf. A080956, A055504.

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

Sequence in context: A218214 A146213 A344311 * 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

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Last modified May 17 16:12 EDT 2022. Contains 353747 sequences. (Running on oeis4.)