

A007569


Number of nodes in regular ngon with all diagonals drawn.
(Formerly M0724)


16



1, 2, 3, 5, 10, 19, 42, 57, 135, 171, 341, 313, 728, 771, 1380, 1393, 2397, 1855, 3895, 3861, 6006, 5963, 8878, 7321, 12675, 12507, 17577, 17277, 23780, 16831, 31496, 30945, 40953, 40291, 52395, 47017, 66082, 65019, 82290
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

I.e., vertex count of the npolygon diagonal intersection graph.  Eric W. Weisstein, Mar 08 2018
Also the circumference of the npolygon diagonal intersection graph (since these graphs are Hamiltonian).  Eric W. Weisstein, Mar 08 2018


REFERENCES

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


LINKS

T. D. Noe, Table of n, a(n) for n=1..1000
Sascha Kurz, mgons in regular ngons
B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135156.
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, 135156 (1998).
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 19952006; arXiv version, which has fewer typos than the SIAM version.
B. Poonen and M. Rubinstein, Mathematica programs for these sequences
M. Rubinstein, Drawings for n=4,5,6,...
Eric Weisstein's World of Mathematics, Graph Circumference
Eric Weisstein's World of Mathematics, Polygon Diagonal Intersection Graph
Eric Weisstein's World of Mathematics, Vertex Count
R. G. Wilson v, Illustration of a(10)
Sequences formed by drawing all diagonals in regular polygon


FORMULA

a(n) = A006561(n)+n.  T. D. Noe, Dec 23 2006


MATHEMATICA

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^270n+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 *)


CROSSREFS

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: A293328 A064236 A245001 * A054317 A065840 A181934
Adjacent sequences: A007566 A007567 A007568 * A007570 A007571 A007572


KEYWORD

easy,nonn,nice,changed


AUTHOR

N. J. A. Sloane, Bjorn Poonen (poonen(AT)math.princeton.edu)


STATUS

approved



