%I A007678 M3411
%S A007678 1,4,11,24,50,80,154,220,375,444,781,952,1456,1696,2500,2466,4029,4500,
%T A007678 6175,6820,9086,9024,12926,13988,17875,19180,24129,21480,31900,33856,
%U A007678 41416,43792,52921,52956,66675,69996,82954,86800,102050
%N A007678 Number of regions in regular n-gon with all diagonals drawn.
%D A007678 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007678 Jean Meeus, Wiskunde Post (Belgium), Vol. 10, 1972, pp. 62-63.
%D A007678 C. A. Pickover, The Mathematics of Oz, Problem 58 "The Beauty of Polygon 
               Slicing", Cambridge University Press, 2002.
%H A007678 T. D. Noe, <a href="b007678.txt">Table of n, a(n) for n=3..1000</a>
%H A007678 Sascha Kurz, <a href="http://www.mathe2.uni-bayreuth.de/sascha/oeis/drawing/
               drawing.html">m-gons in regular n-gons</a>
%H A007678 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 A007678 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 A007678 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 on Discrete 
               Mathematics, pp. 135-6 vol. 11 no.1 1998.
%H A007678 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 A007678 B. Poonen and M. Rubinstein, <a href="http://math.mit.edu/~poonen/papers/
               ngon.m">Mathematica programs for these sequences</a>
%H A007678 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RegularPolygonDivisionbyDiagonals.html">Regular Polygon Division 
               by Diagonals</a>
%H A007678 <a href="Sindx_Pol.html#Poonen">Sequences formed by drawing all diagonals 
               in regular polygon</a>
%F A007678 For odd n>3, a(n) = sumstep {i=5, n, 2, (i-2)*floor(n/2)+(i-4)*ceil(n/
               2)+1} + x*(x+1)*(2*x+1)/6*n), where x=(n-5)/2. Simplifying the floor/
               ceil components gives the PARI code below. - Jon Perry (perry(AT)globalnet.co.uk), 
               Jul 08 2003
%F A007678 Simpler formula for odd n, n>=3: (24 - 42n + 23n^2 - 6n^3 + n^4)/24. 
               - Graeme McRae (g_m(AT)mcraefamily.com), Dec 24 2004
%F A007678 a(n)=A006533(n)-n. - T. D. Noe, Dec 23 2006
%t A007678 del[m_,n_]:=If[Mod[n,m]==0,1,0]; R[n_]:=If[n<3, 0, (n^4-6n^3+23n^2-42n+24)/
               24 + del[2,n](-5n^3+42n^2-40n-48)/48 - del[4,n](3n/4) + del[6,n](-53n^2+310n)/
               12 + del[12,n](49n/2) + del[18,n]*32n + del[24,n]*19n - del[30,n]*36n 
               - del[42,n]*50n - del[60,n]*190n - del[84,n]*78n - del[90,n]*48n 
               - del[120,n]*78n - del[210,n]*48n]; Table[R[n], {n,1,1000}] - T. 
               D. Noe (noe(AT)sspectra.com), Dec 21 2006
%o A007678 (PARI) { a(n)=local(nr,x,fn,cn,fn2); nr=0; fn=floor(n/2); cn=ceil(n/2); 
               fn2=(fn-1)^2-1; nr=fn2*n+fn+(n-2)*fn+cn; x=(n-5)/2; if (x>0,nr+=x*(x+1)*(2*x+1)/
               6*n); nr; }
%Y A007678 Sequences related to chords in a circle: A001006, A054726, A006533, A006561, 
               A006600, A007569, A007678. See also entries for chord diagrams in 
               Index file.
%Y A007678 Sequence in context: A001752 A160860 A143075 this_sequence A159350 A159348 
               A159349
%Y A007678 Adjacent sequences: A007675 A007676 A007677 this_sequence A007679 A007680 
               A007681
%K A007678 easy,nonn,nice
%O A007678 3,2
%A A007678 N. J. A. Sloane (njas(AT)research.att.com), Bjorn Poonen (poonen(AT)math.princeton.edu)
%E A007678 More terms from Graeme McRae (g_m(AT)mcraefamily.com), Dec 26 2004