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A007569 Number of nodes in regular n-gon with all diagonals drawn.
(Formerly M0724)
17
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 n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 08 2018

Also the circumference of the n-polygon 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, 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:math/9508209 [math.MG], 1995-2006; 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^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 *)

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

AUTHOR

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

STATUS

approved

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Last modified May 27 07:54 EDT 2018. Contains 304690 sequences. (Running on oeis4.)