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A006533 Join n equal points around circle in all ways, count regions.
(Formerly M1118)
17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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

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 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,...

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 *)

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

STATUS

approved

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Last modified October 28 02:52 EDT 2020. Contains 338048 sequences. (Running on oeis4.)