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 A007569 Number of nodes in regular n-gon with all diagonals drawn. (Formerly M0724) 17

%I M0724

%S 1,2,3,5,10,19,42,57,135,171,341,313,728,771,1380,1393,2397,1855,3895,

%T 3861,6006,5963,8878,7321,12675,12507,17577,17277,23780,16831,31496,

%U 30945,40953,40291,52395,47017,66082,65019,82290

%N Number of nodes in regular n-gon with all diagonals drawn.

%C I.e., vertex count of the n-polygon diagonal intersection graph. - _Eric W. Weisstein_, Mar 08 2018

%C Also the circumference of the n-polygon diagonal intersection graph (since these graphs are Hamiltonian). - _Eric W. Weisstein_, Mar 08 2018

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A007569/b007569.txt">Table of n, a(n) for n=1..1000</a>

%H Sascha Kurz, <a href="http://www.mathe2.uni-bayreuth.de/sascha/oeis/drawing/drawing.html">m-gons in regular n-gons</a>

%H B. Poonen and M. Rubinstein, <a href="https://doi.org/10.1137/S0895480195281246">Number of Intersection Points Made by the Diagonals of a Regular Polygon</a>, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

%H B. Poonen and M. Rubinstein, <a href="http://math.mit.edu/~poonen/papers/ngon.pdf">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 B. Poonen and M. Rubinstein, <a href="https://arxiv.org/abs/math/9508209">The number of intersection points made by the diagonals of a regular polygon</a>, arXiv:math/9508209 [math.MG], 1995-2006; arXiv version, which has fewer typos than the SIAM version.

%H B. Poonen and M. Rubinstein, <a href="http://math.mit.edu/~poonen/papers/ngon.m">Mathematica programs for these sequences</a>

%H M. Rubinstein, <a href="/A006561/a006561_3.pdf">Drawings for n=4,5,6,...</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCircumference.html">Graph Circumference</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolygonDiagonalIntersectionGraph.html">Polygon Diagonal Intersection Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexCount.html">Vertex Count</a>

%H R. G. Wilson v, <a href="/A006561/a006561_1.pdf">Illustration of a(10)</a>

%H <a href="/index/Pol#Poonen">Sequences formed by drawing all diagonals in regular polygon</a>

%F a(n) = A006561(n)+n. - _T. D. Noe_, Dec 23 2006

%t del[m_,n_]:=If[Mod[n,m]==0,1,0]; Int[n_]:=If[n<4, n, n + 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_, Dec 21 2006 *)

%Y Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.

%K easy,nonn,nice

%O 1,2

%A _N. J. A. Sloane_, Bjorn Poonen (poonen(AT)math.princeton.edu)

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Last modified November 17 10:07 EST 2018. Contains 317275 sequences. (Running on oeis4.)