A006533 Place n equally-spaced points around a circle and join every pair of points by a chord; this divides the circle into a(n) regions. (Formerly M1118)

1, 2, 4, 8, 16, 30, 57, 88, 163, 230, 386, 456, 794, 966, 1471, 1712, 2517, 2484, 4048, 4520, 6196, 6842, 9109, 9048, 12951, 14014, 17902, 19208, 24158, 21510, 31931, 33888, 41449, 43826, 52956, 52992, 66712, 70034, 82993, 86840, 102091, 97776, 124314, 129448, 149986, 155894, 179447, 179280

Offset

1,2

Comments

This sequence and A007678 are two equivalent ways of presenting the same sequence. - N. J. A. Sloane, Jan 23 2020

In contrast to A007678, which only counts the polygons, this sequence also counts the n segments of the circle bounded by the arc of the circle and the straight line, both joining two neighboring points on the circle. Therefore a(n) = A007678(n) + n. - M. F. Hasler, Dec 12 2021

References

Jean Meeus, Wiskunde Post (Belgium), Vol. 10, 1972, pp. 62-63.

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

M. F. Hasler, Interactive illustration of A006561(n) & A006533(n); colored version for n=6 and for n=8.

Sascha Kurz, m-gons in regular n-gons

J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975

Burkard Polster (Mathologer), The hardest "What comes next?" (Euler's pentagonal formula), Youtube video, Oct 17 2020.

Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, No. 1 (1998), pp. 135-156. (Author's copy.).

Bjorn Poonen and Michael Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG]; some typos in the published version are corrected in the revisions from 2006.

Bjorn Poonen and Michael Rubinstein, Mathematica programs for these sequences

Michael Rubinstein, Drawings for n=4,5,6,...

Prasad Balakrishnan Warrier, The physiognomy of the Erdős-Szekeres conjecture (happy ending problem), Math. Student (Indian Math. Soc., 2024) Vol. 93, Nos. 3-4, 28-48.

Sequences formed by drawing all diagonals in regular polygon

Formula

Poonen and Rubinstein give an explicit formula for a(n) (see Mma code).

a(n) = A007678(n) + n. - T. D. Noe, Dec 23 2006

Mathematica

del[m_, n_]:=If[Mod[n, m]==0, 1, 0];
R[n_]:=(n^4-6n^3+23n^2-18n+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, Dec 21 2006 *)

Prog

(Python)
def d(n, m): return not n % m
def A006533(n): return (1176*d(n, 12)*n - 3744*d(n, 120)*n + 1536*d(n, 18)*n - d(n, 2)*(5*n**3 - 42*n**2 + 40*n + 48) - 2304*d(n, 210)*n + 912*d(n, 24)*n - 1728*d(n, 30)*n - 36*d(n, 4)*n - 2400*d(n, 42)*n - 4*d(n, 6)*n*(53*n - 310) - 9120*d(n, 60)*n - 3744*d(n, 84)*n - 2304*d(n, 90)*n + 2*n**4 - 12*n**3 + 46*n**2 - 36*n)//48 + 1 # Chai Wah Wu, Mar 08 2021
(PARI) apply( {A006533(n)=if(n%2, (((n-6)*n+23)*n-18)*n/24+1, ((n^3/2 -17*n^2/4 +22*n -if(n%4, 19, 28) +!(n%6)*(310 -53*n))/12 +!(n%12)*49/2 +!(n%18)*32 +!(n%24)*19 -!(n%30)*36 -!(n%42)*50 -!(n%60)*190 -!(n%84)*78 -!(n%90)*48 -!(n%120)*78 -!(n%210)*48)*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: A164256 A164240 A164214 * A164188 A164186 A164187

Adjacent sequences: A006530 A006531 A006532 * A006534 A006535 A006536

Keyword

nonn,easy,nice

Author

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

Extensions

Added more terms from b-file. - N. J. A. Sloane, Jan 23 2020

Edited definition. - N. J. A. Sloane, Mar 17 2024

Status

approved