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Search: id:A006561
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| A006561 |
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Number of intersections of diagonals in the interior of regular n-gon.
(Formerly M3833)
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+0
17
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| 0, 0, 0, 1, 5, 13, 35, 49, 126, 161, 330, 301, 715, 757, 1365, 1377, 2380, 1837, 3876, 3841, 5985, 5941, 8855, 7297, 12650, 12481, 17550, 17249, 23751, 16801, 31465, 30913, 40920, 40257, 52360, 46981, 66045, 64981, 82251, 80881, 101270
(list; graph; listen)
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OFFSET
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1,5
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv version, which has fewer typos than the SIAM version.
B. Poonen and M. Rubinstein, Mathematica programs for these sequences
B. Poonen & M. Rubinstein, The Number Of Intersection Points Made By The Diagonals Of A Regular Polygon, SIAM Journal in Discrete Mathematics, pp. 135-6 vol. 11 no.1 1998.
Sequences formed by drawing all diagonals in regular polygon
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FORMULA
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For odd n, (n^4-6n^3+11n^2-6n)/24. For even n, use this formula, but then subtract 2 for every 3-crossing, subtract 5 for every 4-crossing, subtract 9 for every 5-crossing, etc. The number to be subtracted is one smaller than a triangular number. - Graeme McRae (g_m(AT)mcraefamily.com), Dec 26 2004
a(n)=A007569(n)-n. - T. D. Noe, Dec 23 2006
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MATHEMATICA
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del[m_, n_]:=If[Mod[n, m]==0, 1, 0]; Int[n_]:=If[n<4, 0, Binomial[n, 4] + del[2, n](-5n^3+45n^2-70n+24)/24 - del[4, n](3n/2) + del[6, n](-45n^2+262n)/6 + del[12, n]*42n + del[18, n]*60n + del[24, n]*35n - del[30, n]*38n - del[42, n]*82n - del[60, n]*330n - del[84, n]*144n - del[90, n]*96n - del[120, n]*144n - del[210, n]*96n]; Table[Int[n], {n, 1, 1000}] - T. D. Noe (noe(AT)sspectra.com), Dec 21 2006
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CROSSREFS
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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: A034509 A034521 A092647 this_sequence A146845 A167710 A126359
Adjacent sequences: A006558 A006559 A006560 this_sequence A006562 A006563 A006564
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KEYWORD
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easy,nonn,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Bjorn Poonen (poonen(AT)math.princeton.edu)
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