

A135565


Number of line segments in regular ngon 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
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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 npolygon 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] (* JeanFrançois 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. AdamsWatters, Feb 23 2008


STATUS

approved



