%I A006561 M3833
%S A006561 0,0,0,1,5,13,35,49,126,161,330,301,715,757,1365,1377,2380,1837,3876,
%T A006561 3841,5985,5941,8855,7297,12650,12481,17550,17249,23751,16801,31465,
%U A006561 30913,40920,40257,52360,46981,66045,64981,82251,80881,101270
%N A006561 Number of intersections of diagonals in the interior of regular n-gon.
%D A006561 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A006561 T. D. Noe, <a href="b006561.txt">Table of n, a(n) for n=1..1000</a>
%H A006561 Sascha Kurz, <a href="http://www.mathe2.uni-bayreuth.de/sascha/oeis/drawing/
               drawing.html">m-gons in regular n-gons</a>
%H A006561 B. Poonen and M. Rubinstein, <a href="http://epubs.siam.org:80/sam-bin/
               dbq/article/28124">Number of Intersection Points Made by the Diagonals 
               of a Regular Polygon</a>, SIAM J. Discrete Mathematics, Vol. 11, 
               pp. 135-156.
%H A006561 B. Poonen and M. Rubinstein, <a href="http://math.mit.edu/~poonen/">The 
               number of intersection points made by the diagonals of a regular 
               polygon</a>, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 
               (1998).
%H A006561 B. Poonen and M. Rubinstein, <a href="http://arXiv.org/pdf/math.MG/9508209">
               The number of intersection points made by the diagonals of a regular 
               polygon</a>, arXiv version, which has fewer typos than the SIAM version.
%H A006561 B. Poonen and M. Rubinstein, <a href="http://math.mit.edu/~poonen/papers/
               ngon.m">Mathematica programs for these sequences</a>
%H A006561 B. Poonen & M. Rubinstein, <a href="http://www.math.uwaterloo.ca/~mrubinst/
               publications/ngon.pdf">The Number Of Intersection Points Made By 
               The Diagonals Of A Regular Polygon</a>, SIAM Journal in Discrete 
               Mathematics, pp. 135-6 vol. 11 no.1 1998.
%H A006561 <a href="Sindx_Pol.html#Poonen">Sequences formed by drawing all diagonals 
               in regular polygon</a>
%F A006561 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
%F A006561 a(n)=A007569(n)-n. - T. D. Noe, Dec 23 2006
%t A006561 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
%Y A006561 Sequences related to chords in a circle: A001006, A054726, A006533, A006561, 
               A006600, A007569, A007678. See also entries for chord diagrams in 
               Index file.
%Y A006561 Sequence in context: A034509 A034521 A092647 this_sequence A146845 A167710 
               A126359
%Y A006561 Adjacent sequences: A006558 A006559 A006560 this_sequence A006562 A006563 
               A006564
%K A006561 easy,nonn,nice
%O A006561 1,5
%A A006561 N. J. A. Sloane (njas(AT)research.att.com), Bjorn Poonen (poonen(AT)math.princeton.edu)