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

1,3

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

Formula

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

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 *)

Crossrefs

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

Sequence in context: A354317 A143785 A182735 * A139488 A028307 A027298

Adjacent sequences: A135562 A135563 A135564 * A135566 A135567 A135568

Keyword

easy,nice,nonn,changed

Author

Franklin T. Adams-Watters, Feb 23 2008

Status

approved