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

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, 80921, 101311, 84883, 123453, 121485

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,...

N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.

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

Robert 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

If n is odd, a(n) = binomial(n,4) + n. - N. J. A. Sloane, Aug 30 2021

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 *)

Prog

(Python)
def d(n, m): return not n % m
def A007569(n): return 2 if n == 2 else n*(42*d(n, 12) - 144*d(n, 120) + 60*d(n, 18) - 96*d(n, 210) + 35*d(n, 24)- 38*d(n, 30) - 82*d(n, 42) - 330*d(n, 60) - 144*d(n, 84) - 96*d(n, 90)) + (n**4 - 6*n**3 + 11*n**2 + 18*n -d(n, 2)*(5*n**3 - 45*n**2 + 70*n - 24) - 36*d(n, 4)*n - 4*d(n, 6)*n*(45*n - 262))//24 # Chai Wah Wu, Mar 08 2021
(PARI) apply( {A007569(n)=A006561(n)+n}, [1..44]) \\ M. F. Hasler, Aug 06 2021

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: A339585 A245001 A365858 * 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