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A135565 Number of line segments in regular n-gon with all diagonals drawn. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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: A224421 A143785 A182735 * A139488 A028307 A027298

Adjacent sequences:  A135562 A135563 A135564 * A135566 A135567 A135568

KEYWORD

easy,nice,nonn

AUTHOR

Franklin T. Adams-Watters, Feb 23 2008

STATUS

approved

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Last modified December 9 19:51 EST 2019. Contains 329879 sequences. (Running on oeis4.)