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 A135565 Number of line segments in regular n-gon with all diagonals drawn. 17
 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 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 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 June 17 22:14 EDT 2021. Contains 345086 sequences. (Running on oeis4.)