A135565 Number of line segments in regular n-gon with all diagonals drawn.

0, 1, 3, 8, 20, 42, 91, 136, 288, 390, 715, 756, 1508, 1722, 2835, 3088, 4896, 4320, 7923, 8360, 12180, 12782, 17963, 16344, 25600, 26494, 35451, 36456, 47908, 38310, 63395, 64800, 82368, 84082, 105315, 99972, 132756, 135014, 165243, 167720

Offset: 1

Author

Franklin T. Adams-Watters, Feb 23 2008

Keywords

• easy
• nice
• nonn

Comments

A line segment (or edge) is considered to end at any vertex where two or more chords meet.

I.e., edge count of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 08 2018

Links

• David W. Wilson, Table of n, a(n) for n = 1..1000
• N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
• Eric Weisstein's World of Mathematics, Edge Count
• Eric Weisstein's World of Mathematics, Polygon Diagonal Intersection Graph
• Sequences formed by drawing all diagonals in regular polygon

Crossrefs

Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.

Programs

Mathematica

del[m_, n_] := Boole[Mod[n, m] == 0];
A007569[n_] :=
If[n < 4, n,
  n + Binomial[n, 4] + del[2, n] (-5 n^3 + 45 n^2 - 70 n + 24)/24 -
   del[4, n] (3 n/2) + del[6, n] (-45 n^2 + 262 n)/6 +
   del[12, n]*42 n + del[18, n]*60 n + del[24, n]*35 n -
   del[30, n]*38 n - del[42, n]*82 n - del[60, n]*330 n -
   del[84, n]*144 n - del[90, n]*96 n - del[120, n]*144 n -
   del[210, n]*96 n];
A007678[n_] :=
  If[n < 3,
   0, (n^4 - 6 n^3 + 23 n^2 - 42 n + 24)/24 +
    del[2, n] (-5 n^3 + 42 n^2 - 40 n - 48)/48 - del[4, n] (3 n/4) +
    del[6, n] (-53 n^2 + 310 n)/12 + del[12, n] (49 n/2) +
    del[18, n]*32 n + del[24, n]*19 n - del[30, n]*36 n -
    del[42, n]*50 n - del[60, n]*190 n - del[84, n]*78 n -
    del[90, n]*48 n - del[120, n]*78 n - del[210, n]*48 n];
a[n_] := A007569[n] + A007678[n] - 1;
Array[a, 40] (* Jean-François Alcover, Sep 07 2017, after Max Alekseyev, using T. D. Noe's code for A007569 and A007678 *)

Formula

a(n) = A007569(n) + A007678(n) - 1. - Max Alekseyev