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A135565
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Number of line segments in regular n-gon with all diagonals drawn.
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34
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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
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1,3
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COMMENTS
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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
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LINKS
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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.
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FORMULA
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MATHEMATICA
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del[m_, n_] := Boole[Mod[n, m] == 0];
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];
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];
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CROSSREFS
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KEYWORD
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easy,nice,nonn
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AUTHOR
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STATUS
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approved
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